Methods and Systems for Determining Fast and Slow Shear Directions in an Anisotropic Formation Using A Logging While Drilling Tool

ABSTRACT

Methods are provided for determining properties of an anisotropic formation (including both fast and slow formations) surrounding a borehole. A logging-while-drilling tool is provided that is moveable through the borehole. The logging-while drilling tool has at least one dipole acoustic source spaced from an array of receivers. During movement of the logging-while-drilling tool, the at least one dipole acoustic source is operated to excite a time-varying pressure field in the anisotropic formation surrounding the borehole. The array of receivers is used to measure waveforms arising from the time-varying pressure field in the anisotropic formation surrounding the borehole. The waveforms are processed to determine a parameter value that represents shear directionality of the anisotropic formation surrounding the borehole.

CROSS-REFERENCE TO RELATED APPLICATION(S)

The subject disclosure claims priority from U.S. Provisional Appl. No.62/322870, filed on Apr. 15, 2016, herein incorporated by reference inits entirety.

TECHNICAL FIELD

The subject disclosure relates to the investigation of earth formations.More particularly, the subject disclosure relates to methods ofmeasuring formation characteristics using logging-while-drilling (LWD)acoustic measurement tools.

BACKGROUND

Wireline borehole acoustic logging is a major part of subsurfaceformation evaluation that is important in oil and gas exploration andproduction. The logging is achieved by lowering a wireline acousticmeasurement tool comprising at least one transmitter and an array ofreceivers into a fluid-filled well, exciting the transmitter(s),recording resulting acoustic waveforms at the receivers, and processingthe recorded waveforms to obtain a depth log of slowness measurements(where slowness is the reciprocal of velocity) along the well. Theacoustic propagation in the borehole is affected by the properties ofrocks surrounding the wellbore. More specifically, the fluid-filledborehole supports propagation of certain number of borehole guided modesthat are generated by a transducer placed inside the borehole fluid.These borehole acoustic modes are characterized by their acousticslowness dispersions which contain valuable information about the rockmechanical properties. Therefore, the acoustic logging can provideanswers pertaining to such properties with diverse applications such asgeophysical calibration of seismic imaging, geomechanical assessment ofwellbore stability, and stress characterization for fracturestimulation. Examples of such acoustic logging are described in i) J. L.A. France, M. A. M. Ortiz, G. S. De, L. Renlie and S. Williams, “Sonicinvestigations in and around the borehole,” Oilfield Review, vol. 18,no. 1, pp. 14-31, March 2006; ii) J. B. U. Haldorsen, D. L. Johnson, T.Plona, B. Sinha, H.-P. Valero and K. Winker, “Borehole acoustic waves,”Oilfield Review, vol. 18, no. 1, pp. 34-43, March 2006; and iii) J.Alford, M. Blyth, E. Tollefsen, J. Crowe, J. Loreto, S. Mohammed, V.Pistre, and A. Rodriguez-Herrera, “Sonic logging while drilling—shearanswers,” Oilfield Review, vol. 24, no. 1, pp. 4-15, January 2012.

Logging-while-drilling (LWD) acoustic tools such as SonicScope 475 andSonicScope 825 of Schlumberger Technology Corporation have beendemonstrated to save a great amount of rig time and to help improve thedrilling efficiency and safety. Processing of the sonic data from theLWD acoustic tools provides monopole compressional and shear slownessesin fast formations and quadrupole shear slowness mostly in slowformations. However, both monopole and quadrupole shear slownessescannot provide a complete anisotropy characterization.

To have a complete anisotropy characterization, one of the mostimportant inputs is the fast-shear azimuthal direction and/or slow shearazimuthal direction, which are desirable for subsequent stress andmechanical analyses of the rock properties around the borehole.

SUMMARY

This summary is provided to introduce a selection of concepts that arefurther described below in the detailed description. This summary is notintended to identify key or essential features of the claimed subjectmatter, nor is it intended to be used as an aid in limiting the scope ofthe claimed subject matter.

Methods are provided for determining properties of an anisotropicformation (including both fast and slow formations) surrounding aborehole. A logging-while-drilling tool is provided that is moveablethrough the borehole. The logging-while drilling tool has at least onedipole acoustic source spaced from an array of receivers. Duringmovement of the logging-while-drilling tool, the at least one dipoleacoustic source is operated to excite a time-varying pressure field inthe anisotropic formation surrounding the borehole. The array ofreceivers are used to measure waveforms arising from the time-varyingpressure field in the anisotropic formation surrounding the borehole.The waveforms are processed to determine shear directionality of theanisotropic formation surrounding the borehole.

Additional aspects, embodiments, objects and advantages of the disclosedmethods may be understood with reference to the following detaileddescription taken in conjunction with the provided drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a schematic diagram of a transversely isotropic formation witha vertical axis of symmetry (TIV).

FIG. 2 is a schematic diagram of a transversely isotropic formation witha horizontal axis of symmetry (TIH).

FIG. 3 is a schematic diagram illustrating a drill-colar mode (dashedcurve labeled “blue”) that propogates in a Logging-While-Drilling (LWD)acoustic measurement tool and that interferes with a formation mode(dashed curve labeled “green”).

FIGS. 4A and 4B are schematic diagrams illustrating cross-dipoleorthogonal firing of a wireline acoustic measurement tool.

FIGS. 5A and 5B are schematic diagrams illustrating non-orthogonaldipole firings of an LWD acoustic measurement tool.

FIG. 6 is a schematic diagram of a wellsite system that can be used inpracticing the embodiments of the subject disclosure.

FIG. 7 is a schematic diagram of a LWD acoustic measurement tool thatcan be used in practicing the embodiments of the subject disclosure.

FIG. 8 is a flowchart illustrating a time-domain workflow according toan embodiment of the subject disclosure.

FIGS. 9A and 9B illustrate synthetic time-domain waveforms arising fromnon-orthogonal D1 and D2 firings, respectively, of a dipole transmitterin the horizontal section of a fast TIV formation.

FIG. 9C illustrates the slowness dispersions of the synthetictime-domain waveforms of FIGS. 9A and 9B in the horizontal section of afast TIV formation.

FIG. 10 illustrates a two-dimensional cost function of thefour-component non-orthogonal LWD waveform rotation for the example ofFIGS. 9A, 9B and 9C.

FIGS. 11A and 11B illustrate rotated time-domain waveforms arising fromnon-orthogonal D1 and D2 firings, respectively, for the example of FIGS.9A, 9B and 9C.

FIG. 11C illustrates the slowness dispersions of the rotated time-domainwaveforms of FIGS. 11A and 11B in the horizontal section of the fast TIVformation.

FIGS. 12A and 12B illustrate synthetic time-domain waveforms arisingfrom non-orthogonal D1 and D2 firings, respectively, from a dipolesource in the horizontal section of a fast TIV formation. The D1 and D2firings are, respectively, 35 and 67 degrees away from the slow shearazimuth.

FIG. 12C illustrates the slowness dispersions of the time-domainwaveforms of FIGS. 12A and 12B in the horizontal section of the fast TIVformation.

FIG. 13 illustrates a two-dimensional cost function of thefour-component non-orthogonal LWD waveform rotation for the example ofFIGS. 12A, 12B and 12C.

FIGS. 14A and 14B illustrate rotated time-domain waveforms arising fromnon-orthogonal D1 and D2 firings, respectively, for the example of FIGS.12A, 12B and 12C.

FIG. 14C illustrates the slowness dispersions of the rotated time-domainwaveforms of FIGS. 14A and 14B in the horizontal section of the fast TIVformation.

FIGS. 15A and 15B illustrate rotated time-domain waveforms arising fromnon-orthogonal D1 and D2 firings, respectively, from a dipole source inthe horizontal section of a slow TIV formation.

FIG. 15C illustrates the slowness dispersions of the rotated time-domainwaveforms of FIGS. 15A and 15B in the horizontal section of the slow TIVformation.

FIG. 16 is a flowchart illustrating a frequency-domain workflowaccording to an embodiment of the subject disclosure.

FIG. 17 is a schematic diagram illustrating shear wave splitting arisingfrom a dipole transmitter in anisotropic formations and principalpolarization directions.

FIG. 18A illustrates an exemplary model slowness dispersion of the fastand slow coupled collar-formation flexural modes arising from a dipolefiring which is 45° away from the fast shear direction in a slowformation. The solid dots between 3.5 and 6 kHz represents a bandlimited dispersion used in the frequency-domain workflow.

FIG. 18B shows an exemplary one-dimensional LWD-DATC cost function,which is constructed using raw inline and crossline waveforms between3.5 and 6 kHz.

FIG. 19A shows two exemplary slowness dispersions of the fast and slowcoupled collar-formation flexural modes arising from a dipole firingwhich is 45° away from the fast shear direction in a slow formation(sameas FIG. 18A), where the slowness dispersions are extracted frompre-rotated inline and crossline waveforms with a pre-determined angleof 60°.

FIG. 19B shows an exemplary one-dimensional LWD-DATC cost functionconstructed from selected slowness dispersions (denoted as solid dots)for the fast and slow flexural modes of FIG. 19A.

FIG. 20A shows two exemplary slowness dispersions of the fast and slowcoupled collar-formation flexural modes arising from a dipole firingthat is 67° away from the fast shear direction (different from FIGS. 18Aand 19A) in a slow formation, where the slowness dispersions areextracted from pre-rotated inline and crossline waveforms with apre-determined angle of 60°.

FIG. 20B shows an exemplary one-dimensional LWD-DATC cost functionconstructed from selected slowness dispersions (denoted as solid dots)for the fast and slow flexural modes of FIG. 20A.

FIGS. 20C shows rotated inline and crossline waveforms for the exampleof FIG. 20A when the inline receivers are parallel to the fast sheardirection of the slow formation.

FIG. 21A shows two exemplary slowness dispersions of the fast and slowcoupled collar-formation flexural modes arising from a dipole firingthat is 85° away from the fast shear direction in a fast formation,where the slowness dispersions are extracted from raw (non-rotated)inline and crossline waveforms

FIG. 21B shows an exemplary one-dimensional LWD-DATC cost functionconstructed from selected slowness dispersions (denoted as solid dots)for the fast and slow flexural modes of FIG. 21A.

FIGS. 21C shows rotated inline and crossline waveforms for the exampleof FIG. 21A when the inline receivers are parallel to the fast sheardirection of the fast formation.

FIG. 22 shows an example computing system that can be used to implementthe time-domain and frequency domain workflows as described herein.

DETAILED DESCRIPTION

The particulars shown herein are by way of example and for purposes ofillustrative discussion of the examples of the subject disclosure onlyand are presented in the cause of providing what is believed to be themost useful and readily understood description of the principles andconceptual aspects of the subject disclosure. In this regard, no attemptis made to show details in more detail than is necessary, thedescription taken with the drawings making apparent to those skilled inthe art how the several forms of the subject disclosure may be embodiedin practice. Furthermore, like reference numbers and designations in thevarious drawings indicate like elements.

As used throughout the specification and claims, the term “downhole”refers to a subterranean environment, particularly in a well orwellbore. “Downhole tool” is used broadly to mean any tool used in asubterranean environment including, but not limited to, a logging tool,an imaging tool, an acoustic tool, a permanent monitoring tool, and acombination tool.

The various techniques disclosed herein may be utilized to facilitateand improve data acquisition and analysis in downhole tools and systems.In this disclosure, downhole tools and systems are provided that utilizearrays of sensing devices that are configured or designed for easyattachment and detachment in downhole sensor tools or modules that aredeployed for purposes of sensing data relating to environmental and toolparameters downhole, within a borehole. The tools and sensing systemsdisclosed herein may effectively sense and store characteristicsrelating to components of downhole tools as well as formation parametersat elevated temperatures and pressures. Chemicals and chemicalproperties of interest in oilfield exploration and development may alsobe measured and stored by the sensing systems contemplated by thepresent disclosure. The sensing systems herein may be incorporated intool systems such as wireline logging tools, measurement-while-drillingand logging-while-drilling tools, permanent monitoring systems, drillbits, drill collars, sondes, among others. For purposes of thisdisclosure, when any one of the terms wireline, cable line, slickline orcoiled tubing or conveyance is used it is understood that any of thereferenced deployment means, or any other suitable equivalent means, maybe used with the present disclosure without departing from the spiritand scope of the present disclosure.

Moreover, inventive aspects lie in less than all features of a singledisclosed embodiment. Thus, the claims following the DetailedDescription are hereby expressly incorporated into this DetailedDescription, with each claim standing on its own as a separateembodiment.

Borehole acoustic logging is a major part of subsurface formationevaluation that is key to oil and gas exploration and production. Thelogging may be achieved using an acoustic measurement tool, whichincludes one or multiple acoustic transducers, or sources, and one ormultiple sensors, or receivers. The acoustic measurement tool may bedeployed in a fluid-field wellbore for purposes of exciting andrecording acoustic waveforms. The receivers thus, may acquire datarepresenting acoustic energy that results from the acoustic energy thatis emitted by the acoustic sources of the acoustic measurement tool.

The acoustic propagation in the borehole is affected by the propertiesof rocks surrounding the wellbore. More specifically, the fluid-filledborehole supports propagation of certain number of borehole guided modesthat are generated by energy from a source that is placed inside theborehole fluid. These borehole acoustic modes are characterized by theiracoustic slowness (i.e., reciprocal of velocity) dispersions, whichcontain valuable information about the rock mechanical properties.Therefore, the acoustic logging may provide answers pertaining to suchdiverse applications as geophysical calibration of seismic imaging,geomechanical assessment of wellbore stability, and stresscharacterization for fracture stimulation. In the context of thisapplication, “acoustic energy” or “sonic energy” refers to energy in thesonic frequency spectrum, and may be, as example, energy between 200Hertz (Hz) and 30 kiloHertz (kHz).

In general, the energy that is emitted by the sources of the acousticmeasurement tool may travel through rock formations as either body wavesor surface waves (called “flexural waves” herein). The body wavesinclude compressional waves, or P-waves, which are waves in which smallparticle vibrations occur in the same direction as the direction inwhich the wave is traveling. The body waves may also include shearwaves, or S-waves, which are waves in which particle motion occurs in adirection that is perpendicular to the direction of wave propagation. Inaddition to the body waves, there are a variety of borehole guided modeswhose propagation characteristics can be analyzed to estimate certainrock properties of the surrounding formation. For instance,axi-symmetric Stoneley and borehole flexural waves are of particularinterest in determining the formation shear slownesses. As describedherein, the flexural waves may also include waves that propagate alongthe acoustic measurement tool.

The acoustic measurement tool may include multiple acoustic sources thatare associated with multiple source classifications, or categories. Forexample, the acoustic measurement tool may include one or multiplemonopole sources. In response to energy from a monopole sonic source,the receivers of the acoustic measurement tool may acquire datarepresenting energy attributable to various wave modes, such as datarepresenting P-waves, S-waves and Stoneley waves.

The acoustic measurement tool may also include one or multipledirectional sources, such as dipole or quadrupole sources, which produceadditional borehole guided waves, which travel through the fluid in theborehole and along the acoustic measurement tool itself Datarepresenting these flexural waves may be processed for such purposes asdetermining the presence or absence of azimuthal anisotropy. Forexample, implementations that are described herein, the datarepresenting the flexural waves is processed for purposes of determininga formation shear slowness.

The speeds at which the aforementioned waves travel are affected byvarious properties of the downhole environment, such as the rockmechanical properties, density and elastic dynamic constants, the amountand type of fluid present in the formation, the makeup of rock grains,the degree of inter-grain cementation and so forth. Therefore, bymeasuring the speed of acoustic wave propagation in the borehole, it ispossible to characterize the surrounding formations based on sensedparameters relating to these properties. The speed, or velocity of agiven sonic wave, or waveform, may be expressed in terms of the inverseof its velocity, which is referred to herein as the “slowness.” In thiscontext, an “acoustic wave” or “acoustic waveform” may refer to aparticular time segment of energy recorded by one or multiple receiversand may correspond to a particular acoustic waveform mode, such as abody wave, flexural or other guided borehole waves.

Certain acoustic waves are non-dispersive, or do not significantly varywith respect to frequency. Other acoustic waves, however, aredispersive, meaning that the wave-slownesses vary as a function offrequency.

The acoustic measurement tool may be deployed on a number of platforms,such as a wireline tool or a logging while drilling (LWD) platform. Inother words, an LWD acoustic measurement tool is disposed on a drillingstring, or pipe. Newly introduced logging-while-drilling (LWD) acousticmeasurement tools have been demonstrated to save a great amount of rigtime and to help drill more efficiently with greater safety margins.Recent progress has enabled LWD acoustic measurement tools to delivercompressional and shear slowness logs in the fast and slow formationsusing monopole and quadrupole transmitters. In this context, a “fastformation” refers to a formation in which the shear wave velocity isgreater than the compressional velocity of the borehole fluid (or“drilling mud”). Otherwise, the formation is a “slow” formation.

However, due to their azimuthal characteristics, the LWD acousticmeasurement tools can be used to obtain only a single reliable shearslowness estimate which is appropriate for isotropic and TIV(transversely isotropic with a vertical axis of symmetry) formations. Anexample TIV formation is shown in FIG. 1. In the TIV formation, elasticproperties are uniform horizontally, but vary vertically.

Currently, there are no reliable techniques for obtaining the fast andslow shear slownesses in anisotropic formations (such as transverselyisotropic with a horizontal axis of symmetry or TIH formations) ororthorhombic formations, because they require the use of directionaldipole firings. For example, see i) J. Alford, M. Blyth, E. Tollefsen,J. Crowe, J. Loreto, S. Mohammed, V. Pistre, and A. Rodriguez-Herrera,“Sonic logging while drilling—shear answers,” Oilfield Review, vol. 24,no. 1, pp. 4-15, January 2012; ii) B. K. Sinha and E. Simsek, “Soniclogging in deviated wellbores in the presence of a drill collar”, 2010SEG Annual Meeting and Exposition, Expanded Abstracts, Denver, Colo.;iii) B. K. Sinha, E. Simsek, and Q-H. Liu, “Elastic wave propagation indeviated wells in anisotropic formations”, Geophysics, 71(6), D191-D202,2006; iv) B. K. Sinha, J. Pabon and C-J. Hsu, “Borehole dipole andquadrupole modes I anisotropic formations”, 2003 IEEE UltrasonicsSymposium Proc., 284-289; and v) B. K. Sinha, E. Simsek, and S.Asvadurov, “Influence of a pipe tool on borehole modes”, Geophysics,vol. 74(3), May-June, 2009. An example TIH formation is shown in FIG. 2.In the TIH formation, elastic properties are uniform in vertical planesparallel to fractures, but vary in the perpendicular horizontaldirection.

In an anisotropic formation, firing of the dipole transmitters generallyexcites both the fast and slow flexural waves, behaving like theshear-wave splitting, with different polarizations and differentvelocities. The shear waves polarized parallel to layering in the TIVformation (e.g., a shale) or vertical fractures in the TIH formation(e.g., a formation with aligned vertical fractures) travel faster thanthe shear waves polarized orthogonal to the layering or fracture.Therefore, the azimuth direction of the fast shear, slow shear andflexural waves can be detected by using firing from cross-dipoletransmitters (e.g., cross-dipole orthogonal firings) in a wirelineacoustic measurement tool. Once the azimuth direction of the fast shearand slow shear is determined, the raw dipole waveforms can be rotated toyield waveforms propagating with the fast and slow shear polarizations.Then the fast and slow shear slownesses can be extracted from theserotated waveforms corresponding to the fast and slow shear azimuthdirections. In summary, both the fast and slow shear slownesses togetherwith the fast shear azimuth direction are inputs to a complete formationanisotropy characterization.

Wireline acoustic measurement tools, such as Schlumberger's SonicScanner™, has been commercialized to provide such complete formationanisotropy characterization. However, the same anisotropycharacterization has not been available in LWD acoustic measurementtools, due to a number of fundamental challenges in LWD acousticmeasurement tools. These challenges include the following:

-   -   strong interference and coupling from collar modes in LWD        acoustic measurement tools;    -   cross-dipole orthogonal firing cannot be maintained in LWD        acoustic measurement tools;    -   tool eccentering in LWD acoustic measurement tools; and    -   signal-to-noise ratio (SNR) in LWD acoustic measurement tools.

With regard to the first challenge regarding strong interference andcoupling from collar modes in the LWD acoustic measurement tool, the LWDacoustic measurement tool has to be mechanically competent for drillingand the drill-collar mode interferes with the formation mode ofinterest. This is unlike the wireline Sonic Scanner™. In fact, thedrill-collar mode dominates the acoustic response especially for thedipole flexural modes in fast formations and strongly couples with theformation mode of interest. Note that a LWD acoustic measurement toolconsists of a stiff drill collar to survive the harsh drillingenvironment. As shown in FIG. 3, the stiff drill-collar supportspropagation of a drill-collar mode (dashed blue curve) that interfereswith a formation flexural mode (dashed green curve). The drill-collarmode intersects the formation flexural mode in a fast formation. Asthese two modes interact in a composite structure, they cannot simplyoverlay on top of each other. Instead, they repel one from the other toform the coupled collar-formation mode (top blue solid curve, alsoreferred to as the tool flexural mode) and the formation-collar mode(bottom green solid curve, also referred to as the formation flexuralmode). Moreover, the drill-collar flexural mode is usually the dominantone. Therefore, a conventional wireline dipole workflow developed forhandling a single formation mode cannot be applied to this complexscenario.

The second challenge stems from the the fact that the cross-dipoleorthogonal firing is unavailable in the LWD acoustic measurement tooldue to tool pipe rotation and therefore cannot be assumed. Thecross-dipole orthogonal firing in a wireline acoustic measurement toolis shown in FIGS. 4A and 4B. The wireline acoustic measurement toolincludes an array of receivers or receiver stations (labeled 1, 2, 3, 4)that are offset at 90 degrees relative to one another about thecircumference of the wireline acoustic measurement tool. FIG. 4A showsthe cross-sectional coordinate system of the X-Dipole firing. TheX-Dipole direction, which is denoted by the solid line with arrow, isoffset θ-degrees from the fast shear direction. This fast sheardirection aligns with the inline receivers (e.g., the azimuthal receiverstations 1 and 3) and perpendicular to the crossline receivers (e.g.,the azimuthal receiver stations 2 and 4) of the receiver array for theX-Dipole firing. The fast and slow shear directions are denoted asdashed black lines with arrows in FIG. 4A. FIG. 4B shows thecross-sectional coordinate system of the Y-Dipole firing. The Y-Dipoledirection, which is denoted by the solid line with arrow,is)(θ+90°)-degree away from the fast shear direction. This fast sheardirection aligns with the inline receivers (e.g., the azimuthal receiverstations 2 and 4) and perpendicular to the crossline receivers (e.g.,the azimuthal receiver stations 1 and 3) for the Y-dipole firing. Thefast and slow shear directions are denoted as dashed black lines witharrows in FIG. 4B. The cross-dipole orthogonal firing (X-dipolefiring/Y-dipole firing) of the wireline acoustic measurement toolenables acquisition of four-component waveforms for the receiver array.The four-component waveforms include an X-Inline waveform, anX-Crossline waveforn, a Y-Inline waveform, and a Y-Crossline waveform.The four-component waveforms can be synthetically rotated towards thefast and slow shear polarization directions by minimizing the totalcrossline energy, for example via the Alford rotation algorithm asdescribed in i) R. M. Alford, “Shear data in the presence of azimuthalanisotropy,” 56th Ann. Internat. Mtg., Sot. Explor. Geophys., ExpandedAbstracts, 476-479, 1986; and ii) C. Esmersoy, K. Koster, M. Williams,A. Boyd and M. Kane, “Dipole shear anisotropy logging”, 64th Ann.Internat. Mtg., Soc. Expl. Geophys., Expanded Abstracts, 1139-1142,1994.

In contrast to the wireline acoustic measurement tool, the orthogonalityof the two LWD dipole firings for each depth is no longer maintained dueto the fast tool rotation with variable speeds during the drillingprocess. An example of the two LWD dipole firings is shown in FIGS. 5Aand 5B. Note that the LWD acoustic measurement tool also includes anarray of receivers or receiver stations (labeled 1, 2, 3, 4) that areoffset at 90 degrees relative to one another about the circumference ofthe LWD acoustic measurement tool. FIG. 5A shows the cross-sectionalcoordinate system of the D1 (Dipole-1) firing. The D1 direction, whichis denoted by the solid line with arrow, is offset θ-degrees from thefast shear direction. This fast shear direction aligns with the inlinereceivers (e.g., the azimuthal receiver stations 1 and 3) andperpendicular to the crossline receivers (e.g., the azimuthal receiverstations 2 and 4) of the receiver array for the D1 firing. The fast andslow shear directions are denoted as dashed black lines with arrows inFIG. 5A. FIG. 5B shows the cross-sectional coordinate system of the D2firing. The D2 direction, which is denoted by the solid line with arrow,is offset (θ+ϕ)-degrees from the fast shear direction. The fast sheardirection aligns with the inline receivers (e.g., the azimuthal receiverstations 2 and 4) and perpendicular to the crossline receivers (e.g.,the azimuthal receiver stations 1 and 3) of the array for the D2 firing.The D1 and D2 dipole firings of the LWD acoustic measurement toolenables acquisition of four-component waveforms for the receiver array.The four-component waveforms include an X-Inline waveform, anX-Crossline waveforn, a Y-Inline waveform, and a Y-Crossline waveform.Due to the non-orthogonal nature of the D1 and D2 firings, thefour-component Alford waveform rotation used for the wireline acousticmeasurement tool cannot be applied to the four-components waveforms ofthe LWD acoustic measurement tool.

The third challenge stems from eccentering of the LWD acousticmeasurement tool during the LWD operations. Since the fast rotation ofdrill string precludes the use of centralizers with the LWD acousticmeasurement tools, stabilizers are used to limit the amount ofeccentering. However, the use of these stabilizers cannot provide acomplete centralization of the LWD acoustic measurement tool. The amountof tool eccentering is further aggravated in deviated wells. Largeamount of eccentering poses additional challenges for the anisotropicprocessing of LWD dipole sonic data.

The fourth challenge stems from the SNR of the LWD acoustic measurementtool. Compared with the wireline logging environment, the acquired datain the LWD environment is usually corrupted by the drilling noise,shocks, and vibration. Furthermore, the directional nature of the dipoletransmitters precludes the use of waveform stacking as used for themonopole and quadrupole logging with a rotating drill string.

To overcome the aforementioned challenges, the present disclosureintroduces two independent workflows that can be used to calculate aparameter value that characterizes the fast shear azimuth direction ofthe formation from the processing of dipole waveforms acquired by anLWD-acoustic measurement tool. The first workflow consists of atime-domain, non-orthogonal waveform rotation algorithm using thefour-component waveforms from the two non-orthogonal LWD dipole firings.The second workflow is based on a frequency-domain processing of thedipole sonic data. This processing is a multi-component rotationalgorithm that accounts for the presence of the drill collar and itsassociated drill-collar flexural mode in the recorded dipole waveforms.The output from either of the two workflows is a parameter value thatrepresents the fast shear azimuth direction of the formation, which canbe used as an input to the formation stress and fracture analyses. Forexample, the parameter value that represents the fast shear azimuthdirection of the formation can be used to synthetically rotate the LWDdipole waveforms acquired by a LWD acoustic measurement tool to pointtoward the fast and slow shear directions. Then the fast and slowformation shear slownesses can be estimated from the rotated waveformsusing a data-driven or model-based and/or workflow. Thus, by cascadingthe LWD dipole waveform rotation and the dipole shear slownessestimation, complete characterization of the anistropy of the formationcan be provided by the LWD tool.

An example of a workflow that estimates fast and slow formation shearslownesses from acoustic waveforms is described in U.S. patentapplication Ser. No. 15/331,946, filed on Oct. 24, 2016, (AttorneyDocket No. IS15.0281-US-NP), entitled “Determining Shear Slowness fromDipole Source-based Measurements Acquired by a Logging-While-DrillingAcoustic Measurement Tool.” In this workflow, for the case where theacoustic measurements are acquired in a fast formation and areassociated with a relatively high SNR (an SNR at, near or above 20 dB,for example), the fast and slow formation shear slownesses can bedetermined from a low frequency formation flexural asymptote engine,which bases the shear slowness determination on a low-frequencyasymptote of extracted flexural dispersions. For the other cases (slowformations or fast formation coupled with a lower SNR), the fast andslow formation shear slownesses can be determined from a model-basedinversion engine, which employs a model that explicitly accounts for thepresence of the acoustic measurement tool in the borehole. As such, themodel-based inversion engine may consider such model inputs ascompressional slowness, formation density, mud density, hole diameter,and so forth. Therefore, for the slow formation, when the tool flexuralacoustic mode significantly interferes with the formation flexuralacoustic mode at the low frequency region, the model-based inversion isused, as the low-frequency asymptote of the extracted flexuraldispersion no longer converges to the shear slowness. In accordance withexample implementations, the model- based inversion engine can use aboundary condition determinant associated with a concentrically placedcylindrical structure to construct the cost function and estimatemultiple physical parameters of interest from numerical optimizationtechniques.

Another example of a workflow that estimates fast and slow formationshear slownesses from acoustic waveforms is described in U.S. patentapplication Ser. No. 15/331,958, filed on Oct. 24, 2016, (AttorneyDocket No. IS15.0282-US-NP), entitled “Determining Shear Slowness Basedon a High Order Formation Flexural Acoustic Mode.” In this workflow,acoustic modes (including the first and third order formation flexuralacoustic modes) are extracted from the acoustic waveforms. The extractedacoustic modes are processed by a high frequency slowness frequencyanalysis (HF-SFA) engine based on input parameters (such as a slownessrange and a frequency range). These ranges may be user selected, inaccordance with example implementations. The HF-SFA enginenon-coherently integrates the dispersion energy along the frequency axisto provide an output. A peak finding engine identifies at least one peakin the integrated energy, and this peak corresponds to an estimated ordetermined shear slowness.

FIG. 6 illustrates a wellsite system in which the workflows of thepresent disclosure can be employed. The wellsite can be onshore oroffshore. In this exemplary system, a borehole 11 is formed insubsurface formations by rotary drilling in a manner that is well known.Embodiments of the present disclosure can also use directional drilling,as will be described hereinafter.

A drill string 12 is suspended within the borehole 11 and has a bottomhole assembly 100 which includes a drill bit 105 at its lower end. Thesurface system includes platform and derrick assembly 10 positioned overthe borehole 11. The assembly 10 includes a rotary table 16, kelly 17,hook 18 and rotary swivel 19. The drill string 12 is rotated by therotary table 16, energized by means not shown, which engages the kelly17 at the upper end of the drill string. The drill string 12 issuspended from a hook 18, attached to a traveling block (also notshown), through the kelly 17 and a rotary swivel 19 which permitsrotation of the drill string relative to the hook. As is well known, atop drive system could alternatively be used.

In the example of this embodiment, the surface system further includesdrilling fluid or mud 26 stored in a pit 27 formed at the well site. Apump 29 delivers the drilling fluid 26 to the interior of the drillstring 12 via a port in the swivel 19, causing the drilling fluid toflow downwardly through the drill string 12 as indicated by thedirectional arrow 8. The drilling fluid exits the drill string 12 viaports in the drill bit 105, and then circulates upwardly through theannulus region between the outside of the drill string and the wall ofthe borehole, as indicated by the directional arrows 9. In this wellknown manner, the drilling fluid lubricates the drill bit 105 andcarries formation cuttings up to the surface as it is returned to thepit 27 for recirculation.

The bottom hole assembly 100 of the illustrated embodiment has alogging-while-drilling (LWD) tool 120, a measuring-while-drilling (MWD)tool 130, a roto-steerable system 150, and drill bit 105. In otherembodiments, the bottom hole assembly 100 can include a mud motor thatis powered by the flow of the drilling fluid and drives the rotation ofthe drill bit 105.

The LWD tool 120 is housed in a special type of drill collar, as isknown in the art, and can contain one or a plurality of known types oflogging tools. It will also be understood that more than one LWD and/orMWD tools can be employed, e.g. as represented at 120A. The LWD toolincludes capabilities for measuring, processing, and storinginformation, as well as for communicating with the surface equipment. Inthe present embodiment, the LWD tool includes at least a dipoletransmitter that transmits directional D1 and D2 dipole firings and anarray of receivers for receiving the four-component waveforms of theacoustic energy that results from the directional D1 and D2 dipolefirings as described herein.

The MWD module 130 is also housed in a special type of drill collar, asis known in the art, and can contain one or more devices for measuringcharacteristics of the drill string and drill bit. In the presentembodiment, the MWD module includes one or more of the following typesof measuring devices: a weight-on-bit measuring device, a torquemeasuring device, a vibration measuring device, a shock measuringdevice, a stick slip measuring device, a direction measuring device, andan inclination measuring device. The LWD tool further includes anapparatus (not shown) for generating electrical power to the downholesystem. This may typically include a mud turbine generator powered bythe flow of the drilling fluid, it being understood that other powerand/or battery systems may be employed.

FIG. 7 schematically illustrates selected components of the acousticmeasurement LWD module 120 of FIG. 6 according to embodiments of thesubject disclosure. A pipe portion 203 defines a mud channel 205.Distributed on the pipe portion 203 is a number of acoustic transmittersincluding a pair of dipole transmitters 201 that transmit directional D1and D2 dipole firings. An array of receivers 207 and receiverelectronics 211 are distributed on the pipe portion 203. The array ofreceivers receive the four-component waveforms of the acoustic energythat results from the directional D1 and D2 dipole firings as describedherein. A surface-located processing facility 151 (FIG. 6) controls theD1 and D2 firings of the dipole transmitters 201 and the receiverelectronics 211. The processing facility 151 can be located in one ormore locations at the wellsite. According to some embodiments, theprocessing facility 151 can process and interpret the data from theacoustic measurement LWD module 120 at one or more locations remote fromthe wellsite. The processing facility 151 may include one or morecentral processing units, storage systems, communications andinput/output modules, a user display, and a user input system.

Time-Domain Workflow

An embodiment of the first workflow that employs a time-domain LWD four-component waveform rotation scheme is summarized in the flow chart ofFIG. 8. The embodiment follows the geometry of the D1 and D2 dipolefiring and the inline and crossline waveforms received by the receiverarray described above with respect to FIGS. 5A and 5B.

In optional block 801, the time domain inline and crossline arraywaveforms of the acoustic energy arising from the D1 LWD dipole firingas received by the receivers of the receiver array can be filtered toremove unwanted noise components. Such filtering can be carried out inone or more domains, such as in the time domain (e.g., time windowprocessing), frequency domain (e.g., bandpass filtering), the slownessdomain (e.g., semblance, Nth-root stacking processing), and in thetime-frequency domain (using wavelets or other time-frequencyrepresentations).

In optional block 803, the time domain inline and crossline arraywaveforms of the acoustic energy arising from the D2 LWD dipole firingas received by the receivers of the receiver array can be filtered toremove unwanted noise components. Such filtering can be carried out inone or more domains, such as in the time domain (e.g., time windowprocessing), frequency domain (e.g., bandpass filtering), the slownessdomain (e.g., semblance, Nth-root stacking processing), and in thetime-frequency domain (using wavelets or other time-frequencyrepresentations).

In block 805, the time-domain inline and crossline array waveformscorresponding to the D1 LWD dipole firing as provided by the waveformfiltering of block 801 (or as received by the receivers of the receiverarray if the waveform filtering of block 801 is omitted) can berepresented as a D1 data vector/matrix u_(D1) as follows:

$\begin{matrix}{{u_{D\; 1}\left( {t,z_{m}} \right)} = \begin{bmatrix}{u_{D\; 1{IN}}\left( {t,z_{m}} \right)} \\{u_{D\; 1{OF}}\left( {t,z_{m}} \right)}\end{bmatrix}} & {{Eqn}.\mspace{11mu} (1)}\end{matrix}$

where u_(D1IN) (t, z_(m)) and u_(D1OF) (t, z_(m)) are, respectively, thetime-doman inline and crossline array waveforms at the m-th receiver ata given time t, and z_(m) denotes the axial location of the m-threceiver.

In total there are M receivers. For example, FIGS. 5A and 5B illustratean example where there are 4 azimuthal receivers at a given axiallocation. Other embodiments can use more than four azimuthal receiversat a given axial location. The azimuthal receiver configuration can bestacked/replicated over a number of axial locations offset from thedipole source for increased sensitivity. In this case, the number ofazimuthal receivers at each axial location can be summed together with asinusoidal function as weights to give the M total receivers.

In an anisotropic formation, the time-domain inline array waveformsu_(D1IN) and crossline array waveforms u_(D1OF) arising from the D1dipole firing consist of contributions from both the fast and slow shearpropagations of the D1 dipole firing. As a result, the D1 datavector/matrix u_(D1) can be represented as follows:

$\begin{matrix}\begin{matrix}{u_{D\; 1} = \begin{bmatrix}{u_{D\; 1{IN}}\left( {t,z_{m}} \right)} \\{u_{D\; 1{OF}}\left( {t,z_{m}} \right)}\end{bmatrix}} \\{= {\begin{bmatrix}{\cos \mspace{11mu} \theta} & {{- \sin}\mspace{11mu} \theta} \\{\sin \mspace{11mu} \theta} & {\cos \mspace{11mu} \theta}\end{bmatrix}\begin{bmatrix}{{s(t)}*{g_{f}\left( {t,z_{m}} \right)}} & 0 \\0 & {{s(t)}*{g_{s}\left( {t,z_{m}} \right)}}\end{bmatrix}}} \\{{\begin{bmatrix}{\cos \mspace{11mu} \theta} & {\sin \mspace{11mu} \theta} \\{{- \sin}\mspace{11mu} \theta} & {\cos \mspace{11mu} \theta}\end{bmatrix}\begin{bmatrix}1 \\0\end{bmatrix}}} \\{\overset{\bigtriangleup}{=}{{R(\theta)}{D\left( {t,z_{m}} \right)}{R^{T}(\theta)}S}}\end{matrix} & {{Eqn}.\mspace{11mu} (2)}\end{matrix}$

Note that the rotation matrix R(θ) is used to project the waveformstwice, one from the D1 dipole source to the fast/slow shear directionsand another (the transpose of the rotation matrix R(θ)) from thefast/slow shear directions to the receiver z_(m). The matrix D(t, z_(m))denotes the propagating waveforms directly in the fast and slow sheardirections (e.g., the diagonal elements). The vector S is a selectionvector for the two-component waveforms for the D1 dipole firing.

In block 807, the time-domain inline and crossline array waveformscorresponding to the D2 LWD dipole firing as provided by the waveformfiltering of block 803 (or as received by the receivers of the receiverarray if the waveform filtering of block 803 is omitted) can berepresented in a D2 data/vector matrix form. Assuming the sourcesignatures from the D1 and D2 dipole firings are the same, thetime-domain inline and crossline array waveforms corresponding to the D2LWD dipole firing can be represented as a D2 data/vector matrix u_(D2)as follows:

$\begin{matrix}\begin{matrix}{u_{D\; 2} = \begin{bmatrix}{u_{D\; 2{IN}}\left( {t,z_{m}} \right)} \\{u_{D\; 2{OF}}\left( {t,z_{m}} \right)}\end{bmatrix}} \\{= \begin{bmatrix}{\cos \; \left( {\theta + \varphi} \right)} & {- {\sin \left( {\theta + \varphi} \right)}} \\{\sin \left( {\theta + \varphi} \right)} & {\cos \left( {\theta + \varphi} \right)}\end{bmatrix}} \\{\begin{bmatrix}{{s(t)}*{g_{f}\left( {t,z_{m}} \right)}} & 0 \\0 & {{s(t)}*{g_{s}\left( {t,z_{m}} \right)}}\end{bmatrix}} \\{{\begin{bmatrix}{\cos \; \left( {\theta + \varphi} \right)} & {\sin \left( {\theta + \varphi} \right)} \\{- {\sin \left( {\theta + \varphi} \right)}} & {\cos \left( {\theta + \varphi} \right)}\end{bmatrix}\begin{bmatrix}1 \\0\end{bmatrix}}} \\{\overset{\bigtriangleup}{=}{{R\left( {\theta + \varphi} \right)}{D\left( {t,z_{m}} \right)}{R^{T}\left( {\theta + \varphi} \right)}S}}\end{matrix} & {{Eqn}.\mspace{11mu} (3)}\end{matrix}$

Note that the rotation matrix R(θ+ϕ) for the D2 dipole firing employs arotation angle of ′+ϕ. Note that the rotation matrix R(θ+ϕ) is used toproject the waveforms twice, one from the D2 dipole source to thefast/slow shear directions and another (the transpose of the rotationmatrix R(θ+ϕ)) from the fast/slow shear directions to the receiverz_(m). The matrix D(t, z_(m)) denotes the propagating waveforms directlyin the fast and slow shear directions (e.g., the diagonal elements). Thevector S is a selection vector for the two-component waveforms for theD2 dipole firing.

It is worth noting that the rotation matrix R(θ+ϕ) has the followingproperty:

R(θ+ϕ)=R(ϕ)R(θ),

R(θ)R ^(T) (θ)=I ₂   Eqn. (4)

where I₂ is the identity matrix of dimension 2.

Therefore, the Eqn. (3) arising from the D2 dipole firing is equivalentto:

U _(D2) =R(ϕ)R(θ)D(t, z _(m))R ^(T)(θ)R ^(T)(ϕ)S   Eqn. (5)

In block 809, a rotation matrix R(ϕ) is defined for the D2 data/vectoru_(D2) as:

$\begin{matrix}{{R(\varphi)} = {\begin{bmatrix}{\cos \mspace{11mu} \varphi} & {{- \sin}\mspace{11mu} \varphi} \\{{- \sin}\mspace{11mu} \varphi} & {\cos \mspace{11mu} \varphi}\end{bmatrix}.}} & {{Eqn}.\mspace{11mu} (6)}\end{matrix}$

Note that the angle ϕ represents the angle difference between the D1 andD2 firings as follows:

$\begin{matrix}{{{{R^{T}(\varphi)}u_{D\; 2}} = {{R(\theta)}{D\left( {t,z_{m}} \right)}{R^{T}(\theta)}\overset{\sim}{S}}}{{{where}\mspace{14mu} \overset{\sim}{S}} = {{{R^{T}(\theta)}S} = {\begin{bmatrix}{\cos \mspace{11mu} \varphi} \\{{- \sin}\mspace{11mu} \varphi}\end{bmatrix}.}}}} & {{Eqn}.\mspace{11mu} (7)}\end{matrix}$

Note that the rotation matrix R^(T) (ϕ) is the transpose of the rotationmatrix R(ϕ)).

In block 811, rotation matrix R(θ) and rotation matrix P(θ) are definedas follows:

$\begin{matrix}{{{{R(\theta)} = \begin{bmatrix}{\cos \mspace{14mu} \theta} & {{- \sin}\mspace{14mu} \theta} \\{{- \sin}\mspace{14mu} \theta} & {\cos \mspace{14mu} \theta}\end{bmatrix}},{and}}{{P(\varphi)} = {\begin{bmatrix}1 & {\cos \mspace{14mu} \varphi} \\1 & {{- \sin}\mspace{14mu} \varphi}\end{bmatrix}.}}} & {{Eqn}.\mspace{14mu} (8)}\end{matrix}$

In block 813, a four-component data vector u can be defined by combiningthe D1 data vector/matrix u_(D1) from the D1 dipole firing (block 805)and the rotated data vectors R^(T) (ϕ)u_(D2)from the D2 dipole firing(block 807 and Eqn. (6)) as follows:

$\begin{matrix}\begin{matrix}{u = {\left\lbrack {u_{D\; 1}\mspace{14mu} {R^{T}(\varphi)}u_{D\; 2}} \right\rbrack = {{R(\theta)}{D\left( {t,z_{m}} \right)}{{R^{T}(\theta)}\left\lbrack {S,\overset{\sim}{S}} \right\rbrack}}}} \\{= {{R(\theta)}{D\left( {t,z_{m}} \right)}{{R^{T}(\theta)}\begin{bmatrix}1 & {\cos \mspace{14mu} \varphi} \\0 & {{- \sin}\mspace{14mu} \varphi}\end{bmatrix}}}} \\{= {{R(\theta)}{D\left( {t,z_{m}} \right)}{R^{T}(\theta)}{P(\varphi)}}}\end{matrix} & {{Eqn}.\mspace{14mu} (9)}\end{matrix}$

Furthermore, the four-component data vector u can be rotated based onthe rotation matrices R(θ) and P(ϕ) (Eqn. (8)) to define a matrix D(t,z_(m)) as follows:

D(t, z _(m))=R ^(T) (θ) [u _(D1) R ^(T)(ϕ)u _(D2) ]P ⁻¹(ϕ)R(θ)   Eqn.(10)

Then, in block 815, a cost function that involves the total crosslineenergy of the matrix D(t, z_(m)) across all of the M receivers and alltime samples can be evaluated and minimized by computer-implementedmethods as follows:

$\begin{matrix}{{\min\limits_{\theta,\varphi}{\sum\limits_{t = t_{0}}^{t_{0} + T}\; {\sum\limits_{m = 1}^{M}\; {{off}\left\{ {D\left( {t,z_{m}} \right)} \right\}}}}} = {\min\limits_{\theta,\varphi}{\sum\limits_{t = t_{0}}^{t_{0} + T}\; {\sum\limits_{m = 1}^{M}\; {{off}\left\{ {{{R^{T}(\theta)}\left\lbrack {u_{D\; 1}\mspace{14mu} {R^{T}(\varphi)}u_{D\; 2}} \right\rbrack}{P^{- 1}(\varphi)}{R(\theta)}} \right\}}}}}} & {{Eqn}.\mspace{14mu} (11)}\end{matrix}$

where off {D} computes the sum of all off-diagonal elements of D, t₀ isthe time index of the first sample, and T is the total sample duration.

In certain scenarios, a magnetometer of the LWD acoustic measurementtool can measure both D1 and D2 firing azimuth directions up to acertain precision with respect to a reference (e.g., the direction ofgravity). Such output can provide an a priori range of θ: θ∈ [θ_(1l),θ_(1h)]) and (θ+ϕ):(θ+ϕ) ∈ [θ_(2l), θ_(2h)] with respect to the samereference direction. This provides a feasible range of the angle ϕ asfollows:

ϕ∈ [θ_(2l)−θ_(1h)−θ_(1l)]  Eqn. (12)

Therefore, the two rotation angles θ, ϕ can be determined bycomputer-implemented methods that solve the following constrainedminimization problem:

$\begin{matrix}{{\min\limits_{\theta,\varphi}{\sum\limits_{t = t_{0}}^{t_{0} + T}\; {\sum\limits_{m = 1}^{M}\; {{off}\left\{ {{{R^{T}(\theta)}\left\lbrack {u_{D\; 1}\mspace{14mu} {R^{T}(\varphi)}u_{D\; 2}} \right\rbrack}{P^{- 1}(\varphi)}{R(\theta)}} \right\}}}}},{s.t.},{\varphi \in \left\lbrack {{\theta_{2l} - \theta_{1h}},{\theta_{2h} - \theta_{1l}}} \right\rbrack}} & {{Eqn}.\mspace{14mu} (13)}\end{matrix}$

Once θ and ϕ have been determined, the fast shear direction of theformation can be calculated as θ₁=θ degrees away from the D1 dipolefiring direction. In other words, the value of the parameter θ₁represents the the fast shear direction of the formation. The fast sheardirection of the formation can also be calculated as θ₂=(θ+ϕ) degreesaway from the D2 dipole firing direction. In other words, the value ofthe parameter 0₂ represents the the fast shear direction of theformation. The slow shear direction of the formation can be calculatedby an offset of 90° relative to the fast shear direction of theformation as is evident from FIGS. 5A and 5B.

Note that if the angle difference between the D1 and D2 firing is known(e.g., the D1 and D2 firing directions can be determined precisely), therotation angle ϕ is known and the two-dimensional minimization of Eqn.(13) reduces to a one-dimensional minimization over θ.

In order to validate the first workflow, consider an example of thehorizontal section of a fast TIV formation (e.g., the Bakken shaleformation). The synthetic data, generated by forward modeling code,simulates a fast formation with an LWD acoustic measurement toolcentered in the borehole. The formation, mud and tool parameters of theforward modeling code are listed in Table 1 below:

TABLE 1 Parameters Values Units DTs(slow)   2170 (141) m/s (us/ft)DTs(fast)   2619 (116) m/s (us/ft) DTc   3473 (88) m/s (us/ft) ρ_(F)  2230 kg/m³ DTm   1500 (203) m/s (us/ft) ρ_(M)   1000 kg/m³ DTs_(tool)  3110 (98) m/s (us/ft) DTc_(tool)   5751 (53) m/s (us/ft) ρ_(T)   7630kg/m³ D 0.1600 (6.3) m (in)

The LWD acoustic measurement tool contains 12 axial receivers placed ata distance from 7 ft to 10.6 ft away from a dipole transmitter with aninter-element spacing of 0.2 ft.

FIGS. 9A and 9B show the synthetic received time-domain array waveformsgenerated from the dipole transmitter in the horizontal section of thefast TIV formation. The D1 and D2 firings are, respectively, 15 and 85degrees away from the slow shear azimuth. In this case, the D1 inlinearray waveforms are mostly dominated by the propagation from the slowshear direction, while the D2 inline array waveforms are mostlydominated by the propagation from the fast shear direction. FIG. 9Cshows the slowness dispersions generated from the dipole transmitter inthe horizontal section of the fast TIV formation. The slownessdispersions represent the dipole flexural dispersion extracted by thematrix pencil method, referred to as the TKO algorithm and described inM. P. Ekstrom, “Dispersion estimation from borehole acoustic arraysusing a modified matrix pencil algorithm,” Proc. 29th Asilomar Conf.Signals, Syst., Comput., vol. 2, Pacific Grove, Calif., November 1995,pp. 449-453. FIG. 9C shows that two flexural modes are present in thefast TIV formation. The upper branch (above 200 us/ft) is the dominantdrill-collar flexural dispersion, while the lower branch is theformation flexural dispersion. The low frequency asymptotes of theformation flexural dispersions (D1 shown with dots labeled with “∘” andD2 shown with dots labeled with “+”) approach to the fast and slow shearslownesses in Table 1. Specifically, the formation flexural dispersionof D1 is similar to the slow flexural dispersion as D1 is close to theslow shear azimuth.

FIG. 10 shows a two-dimensional cost function of the four-componentnon-orthogonal LWD waveform rotation in the (θ₁=θ₂=θ+ϕ) plane (Block 815of FIG. 8) for the synthetic example of FIGS. 9A and 9B. The two dashedlines are constraints for the upper and lower limit of the angledifference between the D1 and D2 firing directions. Such constraints canbe derived from the tolerance of D1 and D2 firing directions as measuredduring rotation of the LWD acoustic measurement tool, for example by amagnometer. It is easy to observe that the global minima are located at(14.2°, 84.9°) and (104.2°, 174.9°) for [θ, θ+ϕ]. Note that there is a90° ambiguity in the [θ, θ+ϕ] plane as one can rotate D1 to the fastshear direction and D2 to the slow shear direction or vice versa. Sincethe workflow values the cost function to find the minimization of thetotal crossline energy, the coordinates of the global minima give theestimated rotation angles. From FIG. 10, the workflow searches for theminima within a bounded region (in between the two dashed lines) and thelocal minima are seen at (01=14.2°, θ₂=84.9°) and (θ₁=104.2°,θ₂=174.9°), where the former one gives the slow shear polarizationdirection (i.e., 14.2° away from the D1 firing or 84.9° away from the D2firing) and the latter one yields the fast shear direction due to a 90°ambiguity. Nevertheless, the 90° ambiguity can be removed by rotatingthe four-component waveforms and identifying which rotated waveformscorrespond to the fast and slow flexural waveforms as described in C.Esmersoy, K. Koster, M. Williams, A. Boyd and M. Kane, “Dipole shearanisotropy logging”, 64th Ann. Internat. Mtg., Soc. Expl. Geophys.,Expanded Abstracts, 1139-1142, 1994.

FIGS. 11A and 11B show the rotated inline and crossline array waveformsfor the D1 and D2 dipole firings of FIGS. 9A and 9B. Particularly, theD1 firing is rotated to the slow shear direction while D2 is rotatedtowards the fast shear direction using the estimated rotation anglesfrom FIG. 10. Note that the crossline energy of the rotated waveforms issignificantly minimized and the inline waveform energy is enhanced. Themodified matrix pencil algorithm (TKO method) can be used to extract thedispersion curves from the rotated inline array waveforms of D1 and D2.FIG. 11C shows the corresponding slowness dispersions. Note thatformation flexural dispersion (dots labeled with “∘”) of the rotated D1captures the slow flexural wave, while the formation flexural dispersion(dots labeled with “+”) of the rotated D2 converges to the fast flexuralshear around 110 us/ft.

FIGS. 12A and 12B show synthetic time-domain array waveforms arisingfrom the D1 and D2 firings of the dipole transmitter in the horizontalsection of a fast TIV formation. The D1 and D2 firings are,respectively, 35° and 67° away from the slow shear azimuth. In thiscase, the D1 and D2 inline array waveforms contain a mixture of both thefast and slow flexural waves. Note that the inline and crossline arraywaveforms for the D1 and D2 dipole firings are closer to an azimuthdirection in between the fast and slow shear direction. Specifically,the the D1 and D2 dipole firings are 35° and 67° away from the slowshear direction, respectively. Compared with the case of FIGS. 9A and9B, more waveform energy is split into the crossline channel, as bothdipole firings move away from either the fast or slow shear directions.FIG. 12C shows the corresponding slowness dispersions. In FIG. 12C, onecan no longer see the formation flexural splitting at low frequenciesfrom the inline receivers.

FIG. 13 shows the two-dimensional cost function (Block 815 of FIG. 8)for the synthetic example of FIGS. 12A and 12B. The two dashed linesrepresent the constraints for the upper and lower limit of the angledifference between the D1 and D2 firing directions. The estimatedrotation angles are (θ₁=34.4°, θ₂=66.7) within the two dashed lines (theconstraints).

FIGS. 14A and 14B show the rotated inline and crossline array waveformsfor the D1 and D2 dipole firings of FIGS. 12A and 12B. Note that thecrossline energy of the rotated array waveforms is significantlyminimized and the inline array waveforms display the fast and slowflexural modes. FIG. 14C shows the corresponding slowness dispersions.Note that the TKO results on the rotated inline array waveforms recoverthe formation flexural dispersions splitting at frequencies below 4 kHz.

FIGS. 15A and 15B shows the rotated synthesized inline and crosslinearray waveforms for D1 and D2 dipole firings in the horizontal sectionof a slow TIV formation. FIG. 15C shows the corresponding slownessdispersions. The TKO results in FIG. 15C show the coupledcollar-flexural dispersions corresponding to the inline waveforms fromthe rotated D1 and D2. The flexural splitting at high frequencies isclearly observed. In this case, the fast and slow shear slownesses canbe inverted from the fast and slow flexural dispersions using amodel-based workflow as described in U.S. patent application Ser. No.15/331,946, filed on Oct. 24, 2016, (Attorney Docket No.IS15.0281-US-NP), entitled “Determining Shear Slowness from DipoleSource-based Measurements Acquired by a Logging-While-Drilling AcousticMeasurement Tool.”

Note that the time-domain workflow as discussed above is computationallyefficient. However, its performance may be affected by the strong noiselevel and the tool eccentering effect in the LWD operation. Moreover,its performance can be affected by mismatch in the source-signatures ofthe D1 and D2 dipole firings. Thus, a complementary frequency-domainworkflow is described below that is intended to relax the limitations ofthe time-domain workflow.

Frequency-Domain Workflow

An embodiment of the second workflow that employs a frequency domain LWDmultiple-component waveform rotation scheme is summarized in the flowchart of FIG. 16. The embodiment follows the geometry of the D1 and D2dipole firing and the inline and crossline waveforms received by thereceiver array described above with respect to FIGS. 5A and 5B. Thefrequency-domain workflow, referred to herein as the LWD-DATC workflow,takes into account the presence of drill collar and its associatedcollar-flexural mode in the waveforms, and outputs a parameter valuethat represents the Fast Shear Azimuth (FSA) direction of the formation.Compared with the time-domain workflow discussed above, the LWD-DATCworkflow is more robust to the presence of noise and any tooleccentering that might be there (insofar as the dispersion is lessaffected). Furthermore, it is based on processing inline and crosslinewaveforms generated by a single directional dipole firing. Thisalleviates the requirement of dipole source matching.

The frequency-domain workflow begins in block 1601 where a fast Fouriertransform is applied to the time domain inline and crossline arraywaveforms of the acoustic energy arising from the either the D1 or D2LWD dipole firing as received by the receivers of the receiver array. Asan optional part of block 1601, the inline and crossline array waveformscan be filtered to remove unwanted noise components. Such filtering canbe carried out in one or more domains, such as in the time domain (e.g.,time window processing), frequency domain (e.g., bandpass filtering),the slowness domain (e.g., semblance, Nth-root stacking processing), andin the time-frequency domain (using wavelets or other time-frequencyrepresentations).

The frequency-domain inline and crossline array waveforms output fromblock 1601 can be represented in a data vector/matrix form. For example,the frequency-domain inline and crossline array waveforms correspondingto the one dipole firing (D1, for instance) can be represented by atwo-component data vector/matrix u_(D1) as follows:

$\begin{matrix}{{{u_{D\; 1}\left( {\omega,z_{m}} \right)} = \begin{bmatrix}{u_{D\; 1{IN}}\left( {\omega,z_{m}} \right)} \\{u_{D\; 1{OF}}\left( {\omega,z_{m}} \right)}\end{bmatrix}},} & {{Eqn}.\mspace{14mu} (14)}\end{matrix}$

where ω is angular frequency.

The two-component data vector/matrix u_(D1) can be expressed as follows:

$\begin{matrix}\begin{matrix}{{u_{D\; 1}\left( {\omega,z_{m}} \right)} = \begin{bmatrix}{u_{D\; 1{IN}}\left( {\omega,z_{m}} \right)} \\{u_{D\; 1{OF}}\left( {\omega,z_{m}} \right)}\end{bmatrix}} \\{{= {{{\begin{bmatrix}{\cos \mspace{14mu} \theta} & {{- \sin}\mspace{14mu} \theta} \\{\sin \mspace{14mu} \theta} & {\cos \mspace{14mu} \theta}\end{bmatrix}\begin{bmatrix}{{s(\omega)}{g_{f}\left( {\omega,z_{m}} \right)}} & 0 \\0 & {{s(\omega)}{g_{s}\left( {\omega,z_{m}} \right)}}\end{bmatrix}}\begin{bmatrix}{\cos \mspace{14mu} \theta} & {\sin \mspace{14mu} \theta} \\{{- \sin}\mspace{14mu} \theta} & {\cos \mspace{14mu} \theta}\end{bmatrix}}\begin{bmatrix}1 \\0\end{bmatrix}}}} \\{{\overset{\Delta}{=}{{R(\theta)}{D\left( {\omega,z_{m}} \right)}{R^{T}(\theta)}S}}}\end{matrix} & {{Eqn}.\mspace{14mu} (15)}\end{matrix}$

This two-component data vector/matrix u_(D1) is a simple transformationof the time-domain data vector of Eqn. (2) into the frequency domain.

In order to model the propogation of the pressure field associated withthe fast and slow flexural waves, consider a dipole transmitter orientedat an angle θ with respect to the fast shear direction as shown in FIG.17. The dipose transmitter waveform can be decomposed into two virtualsources directed along the fast and slow shear directions. The fast andslow flexural waves with corresponding polarization directions propogatealong the borehole in accordance with the fast and slow dispersions. Theinline array waveforms U_(XX) and the crossline array waveforms U_(XY)contain contributions from both the fast and slow flexural waves. Themodel representation of the inline and crossline array waveforms aredenoted by U_(XX) and U_(XY), respectively.

The model representation of the inline and crossline array waveformsU_(XX) and U_(XY) take the following form:

U _(XX) =S _(X) cos² θg_(f) +S _(X) sin² θg_(s),

U _(XY) =S _(X)sin θ cos θg_(f) +S _(X) sin θ θg_(s),   Eqn. (16)

where S_(X) is the source signature for the dipole transmitter alignedalong the X-direction that makes the angle θ with respect to the fastshear direction.

The eigenfunctions for the pressure field associated with the fast andslow flexural waves are given as:

g _(f) (ω, z_(m))=−ρ_(m)ω² cos φζ(k _(r) ^(f) , k _(z) ^(f) , r)

g _(s)(ω, z_(m))=−ρ_(m)ω² cos φζ(k _(r) ^(s) , k _(r) ^(s) , r)   Eqn.(17)

where ρ_(m) is the mud density, r and φ are the radial and azimuthalcoordinates, respectively of a given receiver, k_(r) and k_(z) are theradial and axial wavenumbers, respectively, for the fast and slowflexural waves (denoted by super-script f and s) with the followingassociation

$\begin{matrix}{{\left( k_{r}^{s} \right)^{2} = {\frac{\omega^{2}}{v_{m}^{2}} - \left( k_{z}^{s} \right)^{2}}}{\left( k_{r}^{f} \right)^{2} = {\frac{\omega^{2}}{v_{m}^{2}} - \left( k_{z}^{f} \right)^{2}}}} & {{Eqn}.\mspace{14mu} (18)}\end{matrix}$

with v_(m) denoting the mud compressional velocity.

Note that a function ζ can be used to represent the difference betweenthe LWD acoustic measurement tool and wireline acoustic measurementtool. In the wireline acoustic measurement tool, the tool flexural modeis designed to be significantly slower than the formation flexural modesencountered in logging conditions and is moreover attenuated due to thepresence of an isolation section between the dipole transmitter and thereceivers. Therefore, the formation flexural mode is the dominant onewith no interference from the tool flexural mode. In this case, thefunction ζ is given as:

ζ(k _(r) ^(f) , k _(z) ^(f) , r)=J ₁ (k _(r) ^(f) r)A e ^(jk) ^(z) ^(f)^(x) ^(m) ,

ζ(k _(r) ^(s) , k _(z) ^(s) , r)=J ₁ (k _(r) ^(f) r)A e ^(jk) ^(z) ^(s)^(z) ^(m)   Eqn. (19)

where J₁ is the Bessel function of first kind, and

A is an amplitude coefficient of the Bessel function common to both fastand slow flexural eigenfunctions and obtained from the continuitycondition at the borehole surface for the wireline propagation mode.

The Bessel function J₁ accounts for the formation flexural mode and isdescribed in the following references: i) B. K. Sinha and X. Huang,“Dipole Anisotropy from Two-Component Acquisition: Validation againstsynthetic data,” Schlumberger-Doll Research Note, 1999; ii) B. K. Sinha,S. Bose and X. Huang, “Determination of dipole shear anisotropy of earthformations”, U.S. Pat. No. 6,718,266 B 1 , 2002; and iii) S. Bose, B. K.Sinha, S. Sunaga, T. Endo and H. P. Valero, “Anisotropy processingwithout matched cross-dipole transmitters”, 75th Ann. Internat. Mtg.,Soc. Expl. Geophys., Expanded Abstracts, 2007.

In the LWD acoustic measurement tool there is direct propagation of thestrong drill-collar flexural wave. To account for the existence of therotating tool pipe and the fact that the receivers are located in theannulus just outside of the rotating tool pipe and inside the formation,the function is modified for either a slow formation or a fast formationto include Bessel functions of the second kind in addition to those ofthe first kind.

In a slow formation, the function ζ is modified as follows:

ζ(k _(r) ^(f) , k _(z) ^(f) , r)=[J ₁(k _(r) ^(f) r)A+Y ₁(k _(r) ^(f)r)B]e ^(jk) ^(z) ^(f) ^(x) ^(m) ,

ζ(k _(r) ^(s) , k _(z) ^(s) , r)=[J ₁(k _(r) ^(s) r)A+Y ₁(k _(r) ^(s)r)B]e ^(jk) ^(z) ^(s) ^(z) ^(m) ,   Eqn. (20)

where J₁ is the Bessel function of the first kind,

Y₁ is the Bessel function of the second kind,

k_(r) ^(f) and k_(r) ^(s) are the radial wavenumbers for the fast andslow coupled tool-formation flexural waves, and

A and B are amplitude coefficients for Bessel functions of the first andsecond kind respectively obtained from the continuity conditions at theborehole and tool pipe surface for the LWD propagation mode in the slowformation.

Note that the Y₁ Bessel function of Eqn. (20) is needed to properlyaccount for propagation of the drill-collar flexural wave in the annulusbetween the rotating tool and the formation in this slow formation case.

In the case of a slow formation, Eqn. (15) can be rewritten with thefollowing expression for the two-component data vector/matrix u_(D1):

$\begin{matrix}{{u_{D\; 1}\left( {\omega,z_{m}} \right)} = {\begin{bmatrix}{u_{D\; 1{IN}}\left( {\omega,z_{m}} \right)} \\{u_{D\; 1{OF}}\left( {\omega,z_{m}} \right)}\end{bmatrix} = {{- \rho_{m}}\omega^{2}{s(\omega)}\cos \mspace{14mu} {\phi \left( {{A\begin{bmatrix}{J_{f}\left( z_{m} \right)} & 0 & {J_{s}\left( z_{m} \right)} \\0 & {{J_{s}\left( z_{m} \right)} - {J_{f}\left( z_{m} \right)}} & 0\end{bmatrix}} + {B\begin{bmatrix}{Y_{f}\left( z_{m} \right)} & 0 & {Y_{s}\left( z_{m} \right)} \\0 & {{Y_{s}\left( z_{m} \right)} - {Y_{f}\left( z_{m} \right)}} & 0\end{bmatrix}}} \right)}{\quad\begin{bmatrix}{\cos^{2}\mspace{14mu} \theta} \\{\sin \mspace{14mu} \theta \mspace{14mu} \cos \mspace{14mu} \theta} \\{\sin^{2}\mspace{14mu} \theta}\end{bmatrix}}}}} & {{Eqn}.\mspace{14mu} (21)}\end{matrix}$J _(f)(z _(m))=J ₁(k _(r) ^(f) r)e ^(jk) ^(z) ^(f) ^(z) ^(m) ,

J _(s)(z _(m))=J ₁(k _(r) ^(s) r)e ^(jk) ^(z) ^(s) ^(z) ^(m) ,

where

Y _(f) (z _(m))=Y ₁(k _(r) ^(f) r)e ^(jk) ^(z) ^(f) ^(z) ^(m) , and

Y _(s)(z _(m))=Y ₁(k _(r) ^(s) r)e ^(jk) ^(z) ^(s) ^(z) ^(m) ,   Eqn.(22)

The array waveforms for all of the receivers can be combined into twolong vectors as follows:

$\begin{matrix}{{{u_{D\; 1{IN}}(\omega)} = \begin{bmatrix}{u_{D\; 1{IN}}\left( {\omega,z_{1}} \right)} \\{u_{D\; 1{IN}}\left( {\omega,z_{2}} \right)} \\\vdots \\{u_{D\; 1{IN}}\left( {\omega,z_{M}} \right)}\end{bmatrix}},{{u_{D\; 1{OF}}(\omega)} = {\begin{bmatrix}{u_{D\; 1{OF}}\left( {\omega,z_{1}} \right)} \\{u_{D\; 1{OF}}\left( {\omega,z_{2}} \right)} \\\vdots \\{u_{D\; 1{OF}}\left( {\omega,z_{M}} \right)}\end{bmatrix}.}}} & {{Eqn}.\mspace{14mu} (23)}\end{matrix}$

The vectors of Eqn. (23) can be rewritten into a matrix form as follows:

$\begin{matrix}{\begin{matrix}{{u_{D\; 1}(\omega)} = \begin{bmatrix}{u_{D\; 1{IN}}(\omega)} \\{u_{D\; 1{OF}}(\omega)}\end{bmatrix}} \\{= {{\alpha (\omega)}\left\{ {{A\begin{bmatrix}J_{f} & 0 & J_{s} \\0 & {J_{s} - J_{f}} & 0\end{bmatrix}} + {B\begin{bmatrix}Y_{f} & 0 & Y_{s} \\0 & {Y_{s} - Y_{f}} & 0\end{bmatrix}}} \right){v(\theta)}}} \\{= {{\alpha (\omega)}\left( {{{AJ}(\omega)} + {{BY}(\omega)}} \right){v(\theta)}}}\end{matrix}\mspace{76mu} {where}} & {{Eqn}.\mspace{14mu} (24)} \\{\mspace{76mu} {{{{\alpha (\omega)} = {{- \rho_{m}}\omega^{2}{s(\omega)}\mspace{14mu} \cos \mspace{14mu} \phi}},\mspace{76mu} {{v(\theta)} = \begin{bmatrix}{\cos^{2}\mspace{14mu} \theta} \\{\sin \mspace{14mu} \theta \mspace{14mu} \cos \mspace{14mu} \theta} \\{\sin^{2}\mspace{14mu} \theta}\end{bmatrix}},\mspace{76mu} {J_{f} = \begin{bmatrix}{{J_{1}\left( {k_{r}^{f}r} \right)}e^{{jk}_{z}^{f}z_{1}}} \\{{J_{1}\left( {k_{r}^{j}r} \right)}e^{{jk}_{z}^{f}z_{2}}} \\\vdots \\{{J_{1}\left( {k_{r}^{f}r} \right)}e^{{jk}_{z}^{f}z_{M}}}\end{bmatrix}},\mspace{76mu} {J_{s} = \begin{bmatrix}{{J_{1}\left( {k_{r}^{f}r} \right)}e^{{jk}_{z}^{s}z_{1}}} \\{{J_{1}\left( {k_{r}^{f}r} \right)}e^{{jk}_{z}^{s}z_{2}}} \\\vdots \\{{J_{1}\left( {k_{r}^{f}r} \right)}e^{{jk}_{z}^{s}z_{M}}}\end{bmatrix}},\mspace{76mu} {Y_{f} = \begin{bmatrix}{{Y_{1}\left( {k_{r}^{f}r} \right)}e^{{jk}_{z}^{f}z_{1}}} \\{{Y_{1}\left( {k_{r}^{f}r} \right)}e^{{jk}_{z}^{f}z_{2}}} \\\vdots \\{{Y_{1}\left( {k_{r}^{f}r} \right)}e^{{jk}_{z}^{f}z_{M}}}\end{bmatrix}},{and}}\mspace{76mu} {Y_{s} = {\begin{bmatrix}{{Y_{1}\left( {k_{r}^{f}r} \right)}e^{{jk}_{z}^{s}z_{1}}} \\{{Y_{1}\left( {k_{r}^{f}r} \right)}e^{{jk}_{z}^{s}z_{2}}} \\\vdots \\{{Y_{1}\left( {k_{r}^{f}r} \right)}e^{{jk}_{z}^{s}z_{M}}}\end{bmatrix}.}}}} & {{Eqn}.\mspace{14mu} (25)}\end{matrix}$

The matrices of Eqns. (24) and (25) can be rewritten into a simplifiedform as follows:

$\begin{matrix}{\begin{matrix}{{u_{D\; 1}(\omega)} = \begin{bmatrix}{u_{D\; 1{IN}}(\omega)} \\{u_{D\; 1{OF}}(\omega)}\end{bmatrix}} \\{= {\left\lbrack {{J(\omega)}{v(\theta)}\mspace{14mu} {Y(\omega)}{v(\theta)}} \right\rbrack \begin{bmatrix}{{\alpha (\omega)}A} \\{{\alpha (\omega)}B}\end{bmatrix}}} \\{= {{D\left( {\omega,\theta} \right)}{b(\omega)}}}\end{matrix}.} & {{Eqn}.\mspace{14mu} (26)}\end{matrix}$

In practice, the data vector/matrix u_(D1) of Eqn. (26) is usuallycontaminated by noise as follows:

u _(D1)(ω)=D(ω, θ)b(ω)+n(ω).   Eqn. (27)

To find a solution of the rotation angle θ in a slow formation, anLWD-DATC cost function can be constructed from the data vector/matrixu_(D1) of Eqn. (26). Note that the data vector/matrix u_(D1) of Eqn.(26) is produced from the model of propogation of the pressure fieldassociated with the fast and slow flexural waves of Eqns. (16)-(26). TheLWD-DATC cost function involves the frequency-domain waveforms acrossthe receivers of the array and multiple frequency points as follows:

$\begin{matrix}{\hat{\theta} = {\arg \mspace{14mu} {\max\limits_{\theta}\mspace{14mu} {\sum\limits_{\omega \in \Omega}{{u_{D\; 1}^{H}(\omega)}P_{D{({\omega,\theta})}}{u_{D\; 1}(\omega)}}}}}} & {{Eqn}.\mspace{14mu} (28)}\end{matrix}$

where Ω is the set of selected frequency points,

P_(D(ω, θ)) is the projection matrix onto the subspace of the signalmatrix D(ω, θ) of the form P_(D(ω, θ))=D(ω, θ) (D^(H) (ω, θ)⁻¹ D^(H)(ω,θ), and

D(ω, θ) is a rank-two matrix of the form D(ω, θ)=[J(ω)v(θ) Y(ω)v(θ)] forthe slow formation.   Eqn. (29)

With this cost function, the rotation angle θ is estimated as theparameter that maximizes the total energy projected onto the signalsubspace defined by the two Bessel functions J(ω) and Y(ω) along thefast and slow flexural dispersions. The set of the selected frequencypoints Ω of the cost function is based on estimated dispersion of thefast and slow (tool/formation) flexural modes.

In a fast formation, the function ζ is modified to describe the coupledtool and formation flexural modes as follows:

ζ(k _(r) ^(f) , k _(z) ^(f) , r)=[J ₁(k _(r) ^(f,F) r)A _(F) +Y ₁(k _(r)^(f,F) r)B _(F) ]e ^(jk) ^(z) ^(f,F) ^(z) ^(m) +[J ₁(k _(r) ^(f,T) r)A_(r) +Y ₁(k _(r) ^(f,T) r)B _(T) ]e ^(jk) ^(z) ^(f,T) ^(z) ^(m) ,

(k _(r) ^(f) , k _(z) ^(f) , r)=[J ₁(k _(r) ^(f,F) r)A _(F) +Y ₁(k _(r)^(f,F) r)B _(F) ]e ^(jk) ^(z) ^(f,F) ^(z) ^(m) +[J ₁(k _(r) ^(f,T) r)A_(r) +Y ₁(k _(r) ^(f,T) r)B _(T) ]e ^(jk) ^(z) ^(f,T) ^(z) ^(m) ,   Eqn.(30)

where J₁ is the Bessel function of the first kind, Y₁ is the Besselfunction of the second kind, k_(r) ^(j,F) and k_(r) ^(s,F) are theradial wavenumbers for the fast and slow formation flexural waves, k_(r)^(f,T) and k_(r,s,T) are the radial wavenumbers for the fast and slowtool flexural waves; A_(F) and B_(F) are the amplitude coefficients forthe Bessel function of the first and second kind respectively,representing the LWD formation flexural propagation mode in the fastformation and obtained from the continuity conditions at the boreholeand tool pipe interfaces; and A_(T) and B_(T) are similarly theamplitude coefficients for the Bessel function of the first and secondkind respectively, representing the LWD tool flexural propagation modein the fast formation and obtained from the continuity conditions at theborehole and tool pipe interfaces.

Note that in Eqn. (30) the Y₁ Bessel function of Eqn. (21) is used in asimilar manner as described above with respect to Eqn. (20) to properlyaccount for the propagation of the drill-collar flexural wave in theannulus between the rotating tool and the formation. It also accountsfor the coupling between the rotating tool and the formation.

In the case of a fast formation, Eqn. (15) can be rewritten with thefollowing expression of the two-component data vector/matrix u_(D1):

$\begin{matrix}{\begin{matrix}{{u_{D\; 1}(\omega)} = {{\alpha (\omega)}\begin{pmatrix}{{A_{T}\begin{bmatrix}J_{f}^{T} & 0 & J_{s}^{T} \\0 & {J_{s}^{T} - J_{f}^{T}} & 0\end{bmatrix}} + {B_{T}\begin{bmatrix}Y_{f}^{T} & 0 & Y_{s}^{T} \\0 & {Y_{s}^{T} - Y_{f}^{T}} & 0\end{bmatrix}}} \\{{+ {A_{F}\begin{bmatrix}J_{f}^{F} & 0 & J_{s}^{F} \\0 & {J_{s}^{F} - J_{f}^{F}} & 0\end{bmatrix}}} + {B_{F}\begin{bmatrix}Y_{f}^{F} & 0 & Y_{s}^{F} \\0 & {Y_{s}^{F} - Y_{f}^{F}} & 0\end{bmatrix}}}\end{pmatrix}}} \\{= {{\alpha (\omega)}\left( {{A_{T}{J_{T}(\omega)}} + {B_{T}{Y_{T}(\omega)}} + {A_{F}{J_{f}(\omega)}} + {B_{F}{Y_{F}(\omega)}}} \right){v(\theta)}}}\end{matrix}{where}} & {{Eqn}.\mspace{14mu} (31)} \\{{{\alpha (\omega)} = {{- \rho_{m}}\omega^{2}{s(\omega)}\mspace{14mu} \cos \mspace{14mu} \phi}}{{{v(\theta)} = \begin{bmatrix}{\cos^{2}\mspace{14mu} \theta} \\{\sin \mspace{14mu} \theta \mspace{14mu} \cos \mspace{14mu} \theta} \\{\sin^{2}\mspace{14mu} \theta}\end{bmatrix}},}} & {{Eqn}.\mspace{14mu} (32)} \\{{J_{f}^{T} = \begin{bmatrix}{{J_{1}\left( {k_{r}^{f,T}r} \right)}e^{{jk}_{z}^{f,T}z_{1}}} \\{{J_{1}\left( {k_{r}^{f,T}r} \right)}e^{{jk}_{z}^{f,T}z_{2}}} \\\vdots \\{{J_{1}\left( {k_{r}^{f,T}r} \right)}e^{{jk}_{z}^{f,T}z_{M}}}\end{bmatrix}},} & \square \\{{J_{s}^{T} = \begin{bmatrix}{{J_{1}\left( {k_{r}^{s,T}r} \right)}e^{{jk}_{z}^{s,T}z_{1}}} \\{{J_{1}\left( {k_{r}^{s,T}r} \right)}e^{{jk}_{z}^{s,T}z_{2}}} \\\vdots \\{{J_{1}\left( {k_{r}^{s,T}r} \right)}e^{{jk}_{z}^{s,T}z_{M}}}\end{bmatrix}},} & \square \\{{J_{f}^{F} = \begin{bmatrix}{{J_{1}\left( {k_{r}^{f,F}r} \right)}e^{{jk}_{z}^{f,F}z_{1}}} \\{{J_{1}\left( {k_{r}^{f,T}r} \right)}e^{{jk}_{z}^{f,F}z_{2}}} \\\vdots \\{{J_{1}\left( {k_{r}^{f,T}r} \right)}e^{{jk}_{z}^{s,F}z_{M}}}\end{bmatrix}},} & \square \\{{Y_{s}^{T} = \begin{bmatrix}{{Y_{1}\left( {k_{r}^{s,T}r} \right)}e^{{jk}_{z}^{s,T}z_{1}}} \\{{Y_{1}\left( {k_{r}^{s,T}r} \right)}e^{{jk}_{z}^{s,T}z_{2}}} \\\vdots \\{{Y_{1}\left( {k_{r}^{s,T}r} \right)}e^{{jk}_{z}^{s,T}z_{M}}}\end{bmatrix}},} & \square \\{{Y_{f}^{F} = \begin{bmatrix}{{Y_{1}\left( {k_{r}^{f,F}r} \right)}e^{{jk}_{z}^{f,F}z_{1}}} \\{{Y_{1}\left( {k_{r}^{f,F}r} \right)}e^{{jk}_{z}^{f,F}z_{2}}} \\\vdots \\{{Y_{1}\left( {k_{r}^{f,F}r} \right)}e^{{jk}_{z}^{f,F}z_{M}}}\end{bmatrix}},} & \square \\{Y_{s}^{F} = {\begin{bmatrix}{{Y_{1}\left( {k_{r}^{s,F}r} \right)}e^{{jk}_{z}^{s,F}z_{1}}} \\{{Y_{1}\left( {k_{r}^{s,T}r} \right)}e^{{jk}_{z}^{s,F}z_{2}}} \\\vdots \\{{Y_{1}\left( {k_{r}^{s,T}r} \right)}e^{{jk}_{z}^{s,F}z_{M}}}\end{bmatrix}.}} & \square\end{matrix}$

The matrices of Eqns. (31) and (32) can be rewritten into a simplifiedform as follows:

$\begin{matrix}{\begin{matrix}{{u_{D\; 1}(\omega)} = \begin{bmatrix}{u_{D\; 1{IN}}(\omega)} \\{u_{D\; 1{OF}}(\omega)}\end{bmatrix}} \\{= {\left\lbrack {{J_{T}(\omega)}{v(\theta)}\mspace{14mu} {Y_{T}(\omega)}{v(\theta)}\mspace{14mu} {J_{F}(\omega)}{v(\theta)}\mspace{14mu} {Y_{F}(\omega)}{v(\theta)}} \right\rbrack \begin{bmatrix}{{\alpha (\omega)}A_{T}} \\{{\alpha (\omega)}B_{T}} \\{{\alpha (\omega)}A_{F}} \\{{\alpha (\omega)}B_{F}}\end{bmatrix}}} \\{= {{D\left( {\omega,\theta} \right)}{b(\omega)}}}\end{matrix}.} & {{Eqn}.\mspace{14mu} (33)}\end{matrix}$

This form consists of the same expression as in the case of the slowformation (Eqn. (26)), except that the rank of the matrix D(ω, θ)increases from 2 to 4 due to the separation of the drill-collar andformation flexural modes.

In practice, the data vector/matrix u_(D1) of Eqn. (33) is usuallycontaminated by noise as follows:

u _(D1) (ω)=D(ω, θ)b(ω)+n(ω)   Eqn. (34)

To find a solution of the rotation angle θ in a fast formation, anLWD-DATC cost function can be constructed from the data vector/matrixu_(D1) of Eqn. (33). Note that the data vector/matrix u_(D1) of Eqn.(33) is produced from the model of propogation of the pressure fieldassociated with the fast and slow flexural waves of Eqns. (29)-(33). TheLWD-DATC cost function involves the frequency-domain waveforms acrossall of the receivers of the array and multiple frequency points asfollows:

$\begin{matrix}{\hat{\theta} = {\arg \mspace{14mu} {\max\limits_{\theta}{\sum\limits_{\omega \in \Omega}{{u_{D\; 1}^{H}(\omega)}P_{D{({\omega,\theta})}}{u_{D\; 1}(\omega)}}}}}} & {{Eqn}.\mspace{14mu} (35)}\end{matrix}$

where, again, P_(D(ω, θ)) is the projection matrix onto the subspace ofthe matrix

D(ω, θ), P_(D(ω, θ))=D(ω, θ) (D^(H) (ω, θ)D(ω, θ))⁻¹ D^(H) (ω, θ), and

D(ω, θ) is a rank-four matrix given as

D(ω, θ)=[J _(T)(ω)v(θ) Y _(T)(ω)v(θ) J _(F)(ω)v(θ) Y _(F)(ω)v(θ) ]  Eqn.(36)

In order to construct the LWD-DATC cost function for a slow formation ora fast formation, the frequency-domain workflow rotates the data vectorsoutput from block 1601 (e.g., the two-component data vectors of Eqn.(15)) with a set of one or more predetermined rotation angles in block1603. If the data vectors are rotated by a set of two or morepredetermined rotation angles, the rotated data vectors (which arerotated by the predetermined rotation angle) that shows the largestflexural dispersion splitting are selected for output to block 1605.Note that the set of one or more predetermined rotation angles can beconfigured to cover the fast shear direction based on the fast sheardirections acquired from other depths in the formation.

In block 1605, the rotated data vectors output from block 1603 are usedto estimate the dispersion of fast and slow (tool/formation) flexuralmodes arising from the D1 or D2 LWD dipole firing. Such operations canemploy a variety of methods known in the art for such estimation. Anumber of methods are described in i) U.S. patent application Ser. No.15/331,946, filed on Oct. 24, 2016, (Attorney Docket No. IS15.0281US-NP), entitled “Determining Shear Slowness from Dipole Source-basedMeasurements Acquired by a Logging-While-Drilling Acoustic MeasurementTool;” ii) Ekstrom, M. P., “Dispersion estimation from borehole acousticarrays using a modified matrix pencil algorithm,” 29th Asilomar Conf.Signals, Syst., Comput., Pacific Grove, Calif., pp. 449-453, Oct. 31,1995; iii) Tang, X. M., Li, C., and Patterson, D., “A curve-fittingmethod for analyzing dispersion characteristics of guided elasticwaves,” The 79th SEG Annual Meeting, Houston, Tex., 25-30 Oct. 25-30,2009, pp. 461-465; iv) Wang, C., “Sonic well logging methods andapparatus utilizing parametric inversion dispersive wave processing,”U.S. Pat. No. 7,120,541; v) Aeron, S., Bose, S. and Valero, H.-P.,“Automatic Dispersion Extraction of Multiple Time-Overlapped AcousticSignals,” U.S. Pat. No. 8,339,897; vi) Wang, P. and Bose, S., “Broadbanddispersion extraction of borehole acoustic modes via sparse Bayesianlearning,: 5th IEEE International Workshop on Computational Advances inMulti-Sensor Adaptive Processing, Saint Martine, Dec. 15-18, 2013, pp.268-271; and vii) Wang, P. and Bose, S., “Apparatus for Mode ExtractionUsing Multiple Frequencies,” PCT Publication No. PCT/US14/049703; hereinincorporated by reference in their entireties. The estimated dispersionsfor each of the modes can be represented in terms of the phase slowness,or equivalently, the axial wavenumbers, as a function of frequency,which are then used as described below.

In block 1607, the rotated data vectors output from block 1603 are usedto define a propogation model of the pressure field associated with thefast and slow flexural waves arising from the D1 or D2 LWD dipole firingfor the appropriate fast or slow formation. In one embodiment, for aslow formation, the rotated data vectors output from block 1603 are usedto define the frequency domain waveforms u_(D1IN)(ω) and u_(D1OF)(ω) forthe receivers of the receiver array with respect to the propagationmodel of Eqns. (15)-(26) as described above. The estimated dispersion ofthe fast and slow (tool/formation) flexural modes as computed in block1605 can be used to define the propagation model for the slow formation.For example, the wavenumbers of the fast and slow (tool/formation)flexural modes as computed in block 1605 can be used to compute thematrix D(ω, θ) of Eqn. (29) via the matrices J_(f), J_(s), Y_(f), Y_(s)of Eqn. (25). In another embodiment, for a fast formation, the rotateddata vectors output from block 1603 are used to define the frequencydomain waveforms u_(D1IN)(ω) and u_(D1OF)(ω) for all of the receivers ofthe receiver array with respect to the propagation model of of Eqns.(29)-(32) as described above. The estimated dispersion of the fast andslow (tool/formation) flexural modes as computed in block 1605 can beused to define the propogation model for the fast formation. Forexample, the wavenumbers of the fast and slow (tool/formation) flexuralmodes as computed in block 1605 can be used to compute the matrix D(ω,θ) of Eqn. (36) via the matrices J_(f) ^(T), J_(s) ^(T), J_(f) ^(F),Y_(s) ^(T), Y_(f) ^(F), and Y_(s) ^(F) of Eqn. (32).

In block 1609, the LWD-DATC cost function (e.g., Eqn. (28) for the slowformation or Eqn. (34) for the fast formation) is constructed based onthe propogation model defined in block 1607. The set of the selectedfrequency points Ω of the LWD-DATC cost function is based on estimateddispersion of the fast and slow (tool/formation) flexural modes computedin block 1605. In particular, the frequency points are selected to liein a frequency band where we have sufficiently large separation betweenthe fast and slow flexural (formation/tool) dispersions. In the fastformation case, the selected frequency points can lie in the lowfrequency range where the formation flexural dispersions normally showlarge separation. In the slow formation case, the selected frequencypoints can lie in the relatively high frequency range where thedispersion separation is larger. The LWD-DATC cost function is thenevaluated by computer-implemented methods to determine the angle θ wherethe total energy projected onto the signal subspace defined by the twoBessel functions J(ω) and Y(ω) along the fast and slow flexuraldispersions is maximized.

In block 1611, the angle θ obtained by the maximized cost function inblock 1609 can be used to estimate the fast shear direction of theformation as θ degrees away from the respective D1 or D2 dipole firingdirection. In other words, the parameter value for the angle θ asobtained by the maximized cost function in block 1609 represents thefast shear direction of the formation. The slow shear direction of theformation can be calculated by an offset of 90° relative to the fastshear direction of the formation as is evident from FIGS. 5A and 5B.

In block 1613, the workflow evaluates a stopping criterion to determineif the stopping criterion is satisfied. There are a number of possibleoptions for the choice of stopping criterion. One choice is tore-iterate the process once and see if the estimated fast shear azimuthis close enough to its estimate in the previous iteration. If the twoestimates are close (subject to a threshold), then the workflow ends andoutputs the estimated fast shear azimuth direction in the last iterationor the average value as the final estimate of the fast shear azimuthdirection. If so (yes), the fast shear direction as determined in block1611 is stored and output as the fast shear azimuth direction of theslow formation. If not (no), the data vectors output from block 1601(e.g., the two-component data vectors of Eqn. (15)) can be rotated atone or more predetermined rotation angles in block 1615 in a mannersimilar to block 1603 and the operations of blocks 1605 to 1613 can berepeated for one or more additional iterations until the stoppingcriterion is satisfied.

To evaluate the effectiveness of the LWD-DATC cost function, firstconsider a slow formation, which is the same case as shown in FIGS. 15A,15B and 15C. To construct the LWD-DATC cost function for the slowformation, the wavenumbers of the fast and slow collar-formationflexural modes are used as inputs to compute the matrix D(ω, θ) of Eqn.(29) via the matrices J_(f), J_(s), Y_(f), Y_(s) of Eqn. (25). Tovalidate the proposed frequency-domain workflow, true model wavenumberscan be used to construct the LWD-DATC cost function. In this case, theinline and crossline array waveforms arising from only the D1 firing areused. Specifically, the D1 firing direction is 45° away from the fastshear direction.

FIG. 18A shows the model slowness dispersion of the fast and slowcoupled collar-formation flexural modes. The solid dots between 3.5 and6 kHz represents a band limited dispersion used in the frequency-domainworkflow. FIG. 18B shows the one-dimensional LWD-DATC cost function,which is constructed by using the inline and crossline waveforms between3.5 and 6 kHz from a dipole firing which is 45° away from the fast sheardirection. The maximum of the LWD-DATC cost function corresponds to thefast shear direction, whereas the minimum of the LWD-DATC cost functionrepresents the slow shear direction. The difference between the maximumand minimum reflects the calculated difference between the fast and slowshear polarization directions. Note that the true model slownessdispersion of the fast and slow coupled collar-formation flexural modesbetween 3.5 and 6 kHz is used to construct the LWD-DATC cost function,which computes the projected total signal energy between 3.5 and 6 kHz.It is further seen in FIG. 18B that the maximum of the LWD-DATC costfunction provides the fast shear direction at 46.68°, whereas theminimum yields the slow shear direction. The difference between themaximum and minimum reflects the difference between the fast and slowshear polarization directions.

In practice, the measured dispersion for the fast and slow flexuralwaves may not be known in advance. To address this issue, the inline andcrossline array waveforms can be rotated by a set of pre-determinedangles in block 1603. For instance, the inline and crossline arraywaveforms can be rotated by a set of three pre-determined angles [20°,40°, 60°], and the pre-rotated waveforms showing the largest flexuraldispersion splitting can be selected. Then, in block 1609, estimatedflexural dispersions from the pre-rotated waveforms (block 1605) areused to construct the LWD-DATC cost function (block 1607). In an exampleshown in FIGS. 19A and 19B, two slowness dispersions of the fast andslow coupled collar-formation flexural modes are selected from thepre-rotated waveforms with a pre-determined angle of 60° to constructthe LWD-DATC cost function. The selected slowness dispersions (denotedas solid dots) for the fast and slow flexural modes are used toconstruct and evaluate the LWD-DATC cost function shown in FIG. 19B.Note that the maximum of the LWD-DATC cost function is shifted to theleft to 42.5° and this effect may be attributed to the pre-rotation. Onemay mitigate this effect by using multiple pre-determined angles with afiner step size.

Consider another case with a D2 firing direction that is 67° away fromthe fast- shear direction in a slow formation. Again, the inline andcrossline array waveforms can be rotated by a set of pre-determinedangles in block 1603. For instance, the inline and crossline arraywaveforms can be rotated by a set of three pre-determined angles [20°,40°, 60°], and the pre-rotated waveforms showing the largest flexuraldispersion splitting can be selected. Then, in block 1609, estimatedflexural dispersions from the pre-rotated waveforms (block 1605) areused to construct the LWD-DATC cost function (block 1607). In an exampleshown in FIGS. 20A, 20B and 20C, two slowness dispersions of the fastand slow coupled collar-formation flexural modes are selected from thepre-rotated waveforms with a pre-determined angle of 60° to constructthe LWD-DATC cost function. The selected slowness dispersions (denotedas solid dots) for the fast and slow flexural modes are used toconstruct and evaluate the LWD-DATC cost function shown in FIG. 20B,which attains its maximum at 67.18°. The rotated inline and crosslinewaveforms are shown in FIG. 20C. Note that, the crossline energy is notminimized to zero as the LWD-DATC cost function does not minimize thecrossline energy directly. Nevertheless, the crossline energy isrelatively small when it is compared with the inline waveform energy.

Finally, consider the frequency-domain workflow in a fast formation withthe physical parameters given in Table 1 where a single dipole D2 firingdirection is 85° from the fast shear direction. Again, the inline andcrossline array waveforms can be rotated by a set of pre-determinedangles in block 1603. For instance, the inline and crossline arraywaveforms can be rotated by a set of three pre-determined angles [0°,20°, 40°, 60°], and the pre-rotated waveforms showing the largestflexural dispersion splitting can be selected. Then, in block 1609,estimated flexural dispersions from the pre-rotated waveforms (block1605) are used to construct the LWD-DATC cost function (block 1607). Inan example shown in FIGS. 21A, 21B and 21C, by applying the multiplepre-rotation angles in block 1603, it is found that the raw inline andcrossline array waveforms (rotated by)0° yield the best flexuraldispersion splitting, especially for the formation flexural dispersionat low frequencies. Then, in block 1609, the tool fast and slow flexuraldispersion mode estimates between 3.5 and 6 kHz (the upper branch ofFIG. 21A shows the tool slow flexural mode and the tool fast flexuralmode with dots labelled “slow” and “fast”, respectively) as well as theformation fast and slow flexural dispersion mode estimates between 3.5and 6 kHz (the lower branch of FIG. 21A shows the formation slowflexural mode and the formation fast flexural mode with dots labelled“slow” and “fast”, respectively) are extracted from the raw inline andcrossline array waveforms and used to compute the LWD-DATC cost functionshown in FIG. 20B. The estimated rotation angle is 84.61°. Note that forthe assumed fast formation, the difference between the maximum andminimum of the LWD-DATC cost function is significantly smaller than thatof the slow formation previously considered. In addition, FIG. 21Cdisplays the rotated inline and crossline array waveforms, where theinline array waveforms clearly exhibit significantly larger amplitudeswhile the crossline array waveforms are significantly reduced.

In one aspect, some of the methods and processes described above for thetime-domain and/or the frequency-domain workflows are performed by aprocessor. The term “processor” should not be construed to limit theembodiments disclosed herein to any particular device type or system.The processor may include a computer system which can be part of theLogging and Control System 151 of FIG. 6. The computer system may alsoinclude a computer processor (e.g., a microprocessor, microcontroller,digital signal processor, or general purpose computer) for executing anyof the methods and processes described above. The computer system mayfurther include a memory such as a semiconductor memory device (e.g., aRAM, ROM, PROM, EEPROM, or Flash-Programmable RAM), a magnetic memorydevice (e.g., a diskette or fixed disk), an optical memory device (e.g.,a CD-ROM), a PC card (e.g., PCMCIA card), or other memory device.

Some of the methods and processes described above, can be implemented ascomputer program logic for use with the computer processor. The computerprogram logic may be embodied in various forms, including a source codeform or a computer executable form. Source code may include a series ofcomputer program instructions in a variety of programming languages(e.g., an object code, an assembly language, or a high-level languagesuch as C, C++, or JAVA). Such computer instructions can be stored in anon-transitory computer readable medium (e.g., memory) and executed bythe computer processor. The computer instructions may be distributed inany form as a removable storage medium with accompanying printed orelectronic documentation (e.g., shrink wrapped software), preloaded witha computer system (e.g., on system ROM or fixed disk), or distributedfrom a server or electronic bulletin board over a communication system(e.g., the Internet or World Wide Web).

Alternatively or additionally, the processor may include discreteelectronic components coupled to a printed circuit board, integratedcircuitry (e.g., Application Specific Integrated Circuits (ASIC)),and/or programmable logic devices (e.g., a Field Programmable GateArrays (FPGA)). Any of the methods and processes described above can beimplemented using such logic devices.

FIG. 22 shows an example computing system 300 that can be used toimplement the methods and processes described above for the time-domainand/or the frequency-domain workflows or parts thereof. The computingsystem 300 can be an individual computer system 301A or an arrangementof distributed computer systems. The computer system 301A includes oneor more analysis modules 303 (a program of computer-executableinstructions and associated data) that can be configured to performvarious tasks according to some embodiments, such as the tasks describedabove. To perform these various tasks, an analysis module 303 executeson one or more processors 305, which is (or are) connected to one ormore storage media 307. The processor(s) 305 is (or are) also connectedto a network interface 309 to allow the computer system 301A tocommunicate over a data network 311 with one or more additional computersystems and/or computing systems, such as 301B, 301C, and/or 301D. Notethat computer systems 301B, 301C and/or 301D may or may not share thesame architecture as computer system 301A, and may be located indifferent physical locations.

The processor 305 can include at least a microprocessor,microcontroller, processor module or subsystem, programmable integratedcircuit, programmable gate array, digital signal processor (DSP), oranother control or computing device.

The storage media 307 can be implemented as one or more non-transitorycomputer-readable or machine-readable storage media. Note that while inthe embodiment of FIG. 22, the storage media 307 is depicted as withincomputer system 301A, in some embodiments, storage media 307 may bedistributed within and/or across multiple internal and/or externalenclosures of computing system 301A and/or additional computing systems.Storage media 307 may include one or more different forms of memoryincluding semiconductor memory devices such as dynamic or static randomaccess memories (DRAMs or SRAMs), erasable and programmable read-onlymemories (EPROMs), electrically erasable and programmable read-onlymemories (EEPROMs) and flash memories; magnetic disks such as fixed,floppy and removable disks; other magnetic media including tape; opticalmedia such as compact disks (CDs) or digital video disks (DVDs); orother types of storage devices. Note that the computer-executableinstructions and associated data of the analysis module(s) 303 can beprovided on one computer-readable or machine-readable storage medium ofthe storage media 307, or alternatively, can be provided on multiplecomputer-readable or machine-readable storage media distributed in alarge system having possibly plural nodes. Such computer- readable ormachine-readable storage medium or media is (are) considered to be partof an article (or article of manufacture). An article or article ofmanufacture can refer to any manufactured single component or multiplecomponents. The storage medium or media can be located either in themachine running the machine-readable instructions, or located at aremote site from which machine-readable instructions can be downloadedover a network for execution.

It should be appreciated that computing system 300 is only one exampleof a computing system, and that computing system 300 may have more orfewer components than shown, may combine additional components notdepicted in the embodiment of FIG. 22, and/or computing system 300 mayhave a different configuration or arrangement of the components depictedin FIG. 22. The various components shown in FIG. 22 may be implementedin hardware, software, or a combination of both hardware and software,including one or more signal processing and/or application specificintegrated circuits.

There have been described and illustrated herein several embodiments ofmethods and systems for determining fast and slow shear directions in ananisotropic formation using a logging while drilling tool. Whileparticular embodiments of the invention have been described, thoseskilled in the art will readily appreciate that many modifications arepossible in the examples without materially departing from this subjectdisclosure. For example, the workflows described herein can be adaptedto account for both rotation and sliding motion of the logging whiledrilling tool during excitation of the time-varying pressure field inthe formation surround the borehole and the acquisition of waveformsresulting thereform. Accordingly, all such modifications are intended tobe included within the scope of this disclosure as defined in thefollowing claims.

What is claimed is:
 1. A method of determining properties of ananisotropic formation surrounding a borehole, comprising: providing alogging-while-drilling tool that is moveable through the borehole, thelogging-while drilling tool having at least one dipole acoustic sourcespaced from an array of receivers; during movement of thelogging-while-drilling tool, operating the at least one dipole acousticsource to excite a time-varying pressure field in the anisotropicformation surrounding the borehole; during the movement of thelogging-while-drilling tool, using the array of receivers to measurewaveforms arising from the time-varying pressure field in theanisotropic formation surrounding the borehole; and processing thewaveforms measured by the array of receivers to determine a parametervalue that represents shear directionality of the anisotropic formationsurrounding the borehole.
 2. A method according to claim 1, wherein themovement of the logging-while-drilling tool involves at least one ofrotation and sliding motion of the logging-while-drilling tool.
 3. Amethod according to claim 1, wherein the parameter value represents afast shear direction of the anisotropic formation.
 4. A method accordingto claim 1, wherein the parameter value represents a slow sheardirection of the anisotropic formation.
 5. A method according to claim1, wherein the parameter that represents shear directionality of theanisotropic formation is used to generate synthetically-rotatedwaveforms, and the synthetically-rotated waveforms are used to estimatedipole shear slowness of the formation.
 6. A method according to claim1, wherein: the at least one dipole acoustic source produces first andsecond excitations of an oriented dipole transmitter wavefield in theborehole at different azimuthal directions; and the processing involvesprocessing waveforms measured by the array of receivers in the timedomain for the first and second excitations to evaluate a cost functioninvolving rotation of four-component data vectors.
 7. A method accordingto claim 7, wherein: the four-component data vectors are defined bycombining a data vector arising from the first excitation and a rotateddata vector arising from the second excitation.
 8. A method according toclaim 7, wherein: the data vector arising from the first excitation isdefined by inline and crossline waveforms received by the array ofreceivers and corresponding to the first excitation; and the rotateddata vector arising from the second excitation is defined by inline andcrossline waveforms received by the array of receivers and correspondingto the second excitation.
 9. A method according to claim 6, wherein: thecost function is evaluated to minimize a sum of off-diagonal elementsover a number of time samples and receivers of the receiver array.
 10. Amethod according to claim 6, wherein: the cost function involves totalcrossline energy of a data matrix of rotated four- component datavectors over a number of time samples and receivers of the receiverarray.
 11. A method according to claim 6, wherein: the cost function isconstrained by lower and upper bounds for a difference in azimuthaldirection between the first and second excitations.
 12. A methodaccording to claim 11, wherein: the lower and upper bounds for thedifference in azimuthal direction are determined from output of a sensorof the logging-while-drilling tool that measures azimuthal direction ofthe first and second excitations.
 13. A method according to claim 1,wherein: the at least one dipole acoustic source produces a predefinedexcitation of an oriented dipole transmitter wavefield in the boreholeat a particular azimuthal direction; and the processing involvesprocessing waveforms measured by the array of receivers in the frequencydomain for the predefined excitation to evaluate a cost function whichis based on a propagation model of dispersion extracted from thewaveforms.
 14. A method according to claim 13, wherein: the costfunction involves frequency-domain waveforms for a plurality ofreceivers and multiple frequency points.
 15. A method according to claim13, wherein: the cost function is evaluated to maximize energy projectedonto a signal subspace defined by two Bessel functions J(ω) and Y(ω)along fast and slow flexural dispersions of the waveforms.
 16. A methodaccording to claim 15, wherein: the Bessel function J(ω) is configuredto account for flexural mode of the formation; and the Bessel functionY(ω) is configured to account for propagation of a drill-collar flexuralwave in an annulus between the rotating logging while drilling tool andthe formation as well as coupling between the moving logging-whiledrilling tool and the formation.
 17. A method according to claim 15,wherein: the cost function involves a set of frequency points that areselected based on estimated dispersion of fast and slow flexural modes.18. A method according to claim 13, wherein: the propagation model isdetermined by rotating two-component data vectors over a set of one ormore predetermined rotation angles.
 19. A method according to claim 18,wherein: the two-component data vectors are defined by inline andcrossline waveforms received by the array of receivers that correspondto the predefined excitation.
 20. A method according to claim 18,wherein: the propagation model is determined by rotating two-componentdata vectors over a plurality of predetermined rotation angles, andselecting rotated two-component data vectors that show largest flexuraldispersion splitting.
 21. A method according to claim 20, wherein: theset of one or more predetermined rotation angles is configured to coverfast shear direction of the formation based on fast shear directionsacquired from other depths in the formation.
 22. A method according toclaim 18, wherein: the rotated two-component data vectors are used toestimate dispersion of fast and slow flexural modes arising from thepredefined excitation of the sonic dipole transmitter; and the estimateddispersion of fast and slow flexural modes is used to define thepropagation model.